Number theory - Knowledge

966:"In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad." 4753: 10285: 4999: 1457: 232: 2768: 2563: 5165:: If we count by threes and there is a remainder 2, put down 140. If we count by fives and there is a remainder 3, put down 63. If we count by sevens and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer. If we count by threes and there is a remainder 1, put down 70. If we count by fives and there is a remainder 1, put down 21. If we count by sevens and there is a remainder 1, put down 15. When exceeds 106, the result is obtained by subtracting 105. 9630: 4863:) depend on functions that are known to all, but whose inverses are known only to a chosen few, and would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorized. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems. 11316: 10297: 9590: 2501: 2706: 2317: 11326: 9610: 9600: 9620: 1549: 1136: 47: 2292: 11336: 2013: 10321: 10309: 420: 3861: 2737: 5198:: Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars , 8 the eight winds, and 9 the nine divisions . If the remainder is odd, is male and if the remainder is even, is female. 5724:. This is, in effect, a set of two equations on four variables, since both the real and the imaginary part on each side must match. As a result, we get a surface (two-dimensional) in four-dimensional space. After we choose a convenient hyperplane on which to project the surface (meaning that, say, we choose to ignore the coordinate 3595:(points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is whether there are finitely or infinitely many rational points on a given curve or surface. 2283:. Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and 1803:

as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker. Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof

4878:

set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (i.e., Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one

5119:

the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems

4421:

must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain

680: 2553: 2721:

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.

8804: 2028:, pointed him towards some of Fermat's work on the subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' attention for the subject. Euler's work on number theory includes the following: 4367:

The areas below date from no earlier than the mid-twentieth century, even if they are based on older material. For example, as explained below, algorithms in number theory have a long history, arguably predating the formal concept of proof. However, the modern study of

4783:, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period. 5128:

Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to "work for a living", and would not have belonged to a "leisured middle class") could have been motivated by his own "idle curiosity" in the absence of a "market for new

2626:

and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually goes back to Euler (1730s), who used formal power series and non-rigorous (or implicit) limiting arguments. The use of

1561:(1607–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes. In his notes and letters, he scarcely wrote any proofs—he had no models in the area. 5212:

Perfect and especially amicable numbers are of little or no interest nowadays. The same was not true in medieval times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean (and hence mystical)

2650:

The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of the most interesting questions in each area remain open and are being actively worked on.

4847:

There are two main questions: "Can this be computed?" and "Can it be computed rapidly?" Anyone can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. Fast algorithms for

5338:

of the group. The modern proof would have been within Fermat's means (and was indeed given later by Euler), even though the modern concept of a group came long after Fermat or Euler. (It helps to know that inverses exist modulo

4028:

defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables; that is, four dimensions). The number of (doughnut) holes in the surface is called the

3876:

The rephrasing of questions on equations in terms of points on curves is felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve (that is, rational or integer solutions to an equation

1418:

may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the

4895:

said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations". Elementary number theory is taught in

820:). By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to 9124:). Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal. Other popular first introductions are: 8614: 7284: 6476:, pp. 48, 53–54. The initial subjects of Fermat's correspondence included divisors ("aliquot parts") and many subjects outside number theory; see the list in the letter from Fermat to Roberval, 22.IX.1636, 1407:; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. 5153:

Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things.

550: 4879:

solution, the solution itself provides a proof of the fact that a solution exists. It cannot be proven that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.)

9052: 1047: 2331:

and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied

7075:, section 1: "The main difference is that in algebraic number theory one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory one looks for 3065:. Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study. 3340: 1527:—who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late 31: 6945:. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work does not really belong to " (quoted in 5868: 4252: 3196:

were introduced; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and

3051: 1397: 1347: 5049:

had to explain: "By arithmetic, Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers." (

4569: 1651: 1296:, it seems to be the case that Indian mathematics is otherwise an indigenous tradition; in particular, there is no evidence that Euclid's Elements reached India before the 18th century. 935:

Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early

3550:, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields. 5722: 2461: 5392: 4101:, determine how well it can be approximated by rational numbers. One seeks approximations that are good relative to the amount of space required to write the rational number: call 758:; it is very simple material ("odd times even is even", "if an odd number measures an even number, then it also measures half of it"), but it is all that is needed to prove that 5280: 2119: 5573:

can be expressed in terms of the four basic operations together with square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equation

4394:. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite. 3741: 2176: 1975: 1915: 1862: 528: 3648: 3446: 3413: 3115: 2813:, are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory. 1801: 1272: 5638: 3855: 2379: 2083: 8710: 3221: 2887:, a key analytic object at the roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a 2532:

The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.

2271: 1218: 8608: 7443: 4168: 1745: 1705: 814: 781: 6133:

The date of the text has been narrowed down to 220–420 CE (Yan Dunjie) or 280–473 CE (Wang Ling) through internal evidence (= taxation systems assumed in the text). See

3273: 4794:, together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation 4072: 4026: 3985: 3916: 4830: 2212:

of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of

2001: 475: 2946: 3809: 3775: 3680: 6065:, p. 212: "Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit;..." 5419:

Up to the second half of the seventeenth century, academic positions were very rare, and most mathematicians and scientists earned their living in some other way (

4422:

sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.

3181:

amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.

4696: 4127: 2487: 715: 4736: 4716: 4662: 4622: 4589: 4492: 4472: 4419: 4312: 4292: 4272: 4188: 4099: 3936: 3380: 3360: 3241: 3175: 3155: 3135: 2966: 1434:

Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by

1917:

has no non-trivial solutions, and that this could also be proven by infinite descent. The first known proof is due to Euler (1753; indeed by infinite descent).

3575:

For example, an equation in two variables defines a curve in the plane. More generally, an equation or system of equations in two or more variables defines a

5088:, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." ( 9056: 8820: 3068:

It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in

1584:, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day. 4928:(FFT) algorithm, which is used to efficiently compute the discrete Fourier transform, has important applications in signal processing and data analysis. 1172:

is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form

724:, for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems. 8062: 2619: 2392: 4628:, together with some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with 8976: 8424: 4315: 3448:

are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by

952:

While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition.

2463:

and worked on quadratic forms along the lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for

2295:"Here was a problem, that I, a ten-year-old, could understand, and I knew from that moment that I would never let it go. I had to solve it." —Sir 3495:.) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions 4891:(1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory. In 1974, 4866:

Some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven, as a solution to

4390:

Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually

1119:; its solution (absent from the manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed 9745: 9666: 9120:

Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods (

7351: 5770:

is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".

2867:, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define 8191: 1531:, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' 259: 7952: 7491: 5728:), we can plot the resulting projection, which is a surface in ordinary three-dimensional space. It then becomes clear that the result is a 5585:

in the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)

675:{\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right)\right)^{2}+1=\left({\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)^{2},} 6855: 4314:

cannot be approximated well, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of

2300: 1162:; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's 5459:

usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy (

9707: 8662: 7171: 6186: 4318:) are critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in 882:(3rd, 4th or 5th century CE). (There is one important step glossed over in Sunzi's solution: it is the problem that was later solved by 10359: 4918:: Public-key encryption schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors. 1076:

to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's

9613: 8706:

Alberuni's India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy and Astrology of India, Vol. 1

7873:"Methods and Traditions of Babylonian Mathematics: Plimpton 322, Pythagorean Triples and the Babylonian Triangle Parameter Equations" 8704: 2643:(1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory ( 9712: 8103: 5080: 921:

There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere.

301:(1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study 17: 8459: 4786:

An early case is that of what is now called the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the

970:

Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim:

9313: 2516:

and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation (

1002: 11069: 11041: 8018: 3572:

has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.

910: 11094: 9261: 9225: 9182: 9151: 9092: 9024: 8918: 8887: 8852: 8743: 8597: 8489: 8408: 8377: 8324: 8272: 8219: 8172: 8117: 8089: 8032: 7986: 7931: 7907: 7840: 7783: 7707: 7638: 7613: 7582: 7485: 7437: 7316: 7278: 7248: 7223: 7183: 5410:, p. 7). Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm. 1747:. These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the 8700: 6511: 10945: 10108: 8159:

Hopkins, J.F.P. (1990). "Geographical and Navigational Literature". In Young, M.J.L.; Latham, J.D.; Serjeant, R.B. (eds.).

5476:

Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance,

5327: 5187:

Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child.

1707:

is not divisible by any prime congruent to −1 modulo 4; and every prime congruent to 1 modulo 4 can be written in the form

195: 6425: 5403: 2891:(here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function. 1152: 11099: 10378: 8879: 5027: 3278: 10611: 10253: 9738: 9659: 8243: 7601: 7330: 6505: 5467:, p. 9). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions. 4355:

is arguably used most often when one wishes to emphasize the connections to modern algebraic geometry (for example, in

2805:

in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.

2201: 8214:. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society. 4962:

into 12 equal parts. This has been studied using number theory and in particular the properties of the 12th root of 2.

2327:(1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the 393:

regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular,

11251: 11079: 10616: 10325: 7898:

von Fritz, Kurt (2004). "The Discovery of Incommensurability by Hippasus of Metapontum". In Christianidis, J. (ed.).

5557:). The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started with 2328: 252: 59: 4938:

has connections to the distribution of prime numbers and has been studied for its potential implications in physics.

1587:

In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.

11339: 10440: 9789: 7243:"Applications of number theory to numerical analysis", Lo-keng Hua, Luogeng Hua, Yuan Wang, Springer-Verlag, 1981, 5789: 4852:

are now known, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.

4668:, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of 4193: 2230:. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form 7104:, section 3: " defined what we now call the Riemann zeta function Riemann's deep work gave birth to our subject " 5732:, loosely speaking, the surface of a doughnut (somewhat stretched). A doughnut has one hole; hence the genus is 1. 4944:: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes. 2971: 1352: 1302: 727:

While evidence of Babylonian number theory is only survived by the Plimpton 322 tablet, some authors asserts that

10727: 10203: 8994: 8359: 7767: 7308:

Digital Signal Processing Algorithms : Number Theory, Convolution, Fast Fourier Transforms, and Applications

4497: 4373: 2829: 2640: 2220:. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the 1604: 5447:(1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 ( 3811:

to the former. It is also the same as asking for all points with rational coordinates on the curve described by

1410:

Brahmagupta (628 AD) started the systematic study of indefinite quadratic equations—in particular, the misnamed

11018: 10980: 10644: 10352: 10301: 9344: 7793: 2875:

in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question

2566: 2490: 1116: 1054: 6183:"Eusebius of Caesarea: Praeparatio Evangelica (Preparation for the Gospel). Tr. E.H. Gifford (1903) – Book 10" 11160: 11137: 10867: 10857: 9731: 9652: 9253: 8910: 7652: 7175: 5651: 4972: 2387:(1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the 2121:; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by 1447: 8527: 8056: 5899:, pp. 36–40) discusses the table in detail and mentions in passing Euclid's method in modern notation ( 5525:) that send elements of L to other elements of L while leaving all elements of K fixed. Thus, for instance, 3868:, that is, a curve of genus 1 having at least one rational point. (Either graph can be seen as a slice of a 2398: 977:

Plato had a keen interest in mathematics, and distinguished clearly between arithmetic and calculation. (By

11241: 10829: 10737: 10649: 10425: 10410: 10313: 9603: 9390: 8800: 7151: 5358: 4319: 2209: 915: 245: 8547: 5249: 2088: 1427:(cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in 11329: 11064: 10569: 10228: 9784: 9385: 9370: 9306: 9281: 9108: 8826: 8587: 8340: 7728: 5444: 5323: 4747: 4398: 4385: 4343:, which is a collection of graphical methods for answering certain questions in algebraic number theory. 2760: 541:

The table's layout suggests that it was constructed by means of what amounts, in modern language, to the

88: 8935: 427:

The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet

11301: 10950: 9799: 8316: 8164: 8024: 5440: 5218: 4867: 4078: 3693: 2876: 2691: 2128: 2036: 1927: 1867: 1814: 1590: 728: 480: 7722: 7162:(1976). "Hilbert's Tenth Problem: Diophantine Equations: Positive Aspects of a Negative Solution". In 11319: 11246: 11221: 11084: 10732: 10345: 10213: 10185: 9822: 9565: 9524: 9403: 8770: 8511: 4833: 4333: 2520:) and devoted a section to computational matters, including primality tests. The last section of the 1748: 1080:). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the 930: 873: 381: 341: 210: 200: 190: 7647:

Apostol, Tom M. (1981). "An Introduction to the Theory of Numbers (Review of Hardy & Wright.)".

7234:

Computer science and its relation to mathematics" DE Knuth – The American Mathematical Monthly, 1974

4430: 3604: 3418: 3385: 3087: 1757: 1507:, 953 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary 1223: 332:) that encode properties of the integers, primes or other number-theoretic objects in some fashion ( 11170: 11003: 10596: 10465: 10258: 9409: 8955: 5597: 5012: 4442: 4391: 3814: 2678:

was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by

2338: 2304: 2042: 1921: 1424: 1159: 1140: 1100: 689:

exercises. If some other method was used, the triples were first constructed and then reordered by

8183: 5870:. Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.( 3199: 2233: 1175: 11231: 11165: 11056: 10872: 10539: 10143: 10133: 10103: 10037: 9772: 9629: 9352: 9336: 9084: 9002: 8109: 7945: 7742: 6847: 4787: 4132: 3177:

is a fixed rational number whose square root is not rational.) For that matter, the 11th-century

2900: 2541: 1710: 1670: 1435: 795: 762: 686: 3246: 2194:, first misnamed by Euler. He wrote on the link between continued fractions and Pell's equation. 11360: 11296: 11127: 11008: 10775: 10765: 10760: 10241: 10138: 10118: 10113: 10042: 9767: 9697: 9593: 9413: 9362: 9299: 8950: 8293: 8077: 8052: 8048: 7464:

Livné, R. (2001), Ciliberto, Ciro; Hirzebruch, Friedrich; Miranda, Rick; Teicher, Mina (eds.),

5952: 4925: 4600: 4446: 3055: 2880: 2731: 2599: 1864:

has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that

1107:

he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by

865:, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the 821: 542: 333: 205: 180: 83: 9633: 9048: 8685: 8006: 6182: 4947:

Communications: The design of cellular telephone networks requires knowledge of the theory of

4036: 3990: 3949: 3880: 11266: 11236: 11226: 11122: 11036: 10912: 10852: 10819: 10809: 10699: 10664: 10654: 10591: 10460: 10435: 10430: 10395: 10268: 10198: 10075: 9999: 9938: 9923: 9918: 9895: 9777: 9692: 9570: 9499: 8398: 7511:

Cartwright, Julyan H. E.; Gonzalez, Diego L.; Piro, Oreste; Stanzial, Domenico (2002-03-01).

6562:, Chap. II. The standard Tannery & Henry work includes a revision of Fermat's posthumous 4941: 4875: 4837: 4797: 4769: 4356: 4323: 3453: 2884: 2825: 2740: 2636: 2513: 2384: 2324: 2274: 2221: 2122: 1980: 1488: 990: 720:

It is not known what these applications may have been, or whether there could have been any;

442: 362: 329: 215: 129: 3452:) seems to have come from the study of higher reciprocity laws, that is, generalisations of 2916: 11026: 10998: 10970: 10965: 10794: 10770: 10722: 10707: 10689: 10679: 10674: 10636: 10586: 10581: 10498: 10444: 10248: 10128: 10123: 10047: 9948: 9560: 9395: 9235: 9143: 8127: 7917: 7660: 7384: 5335: 5331: 5214: 4897: 3780: 3746: 3653: 3569: 3559: 2817: 2675: 2572:

Starting early in the nineteenth century, the following developments gradually took place:

2388: 1276: 1089: 1066: 1062: 1053:

was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of

1050: 751: 721: 321: 298: 175: 134: 103: 9623: 7631:

Chinese Mathematics in the Thirteenth Century: the "Shu-shu Chiu-chang" of Ch'in Chiu-shao

7193: 4981:. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the 4624:

of arithmetic significance, such as the primes or the squares) and, arguably, some of the

2024:(1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur 1483:

ordered translations of many Greek mathematical works and at least one Sanskrit work (the

8: 11291: 11216: 11132: 11117: 10882: 10669: 10626: 10621: 10518: 10508: 10480: 10263: 10173: 10095: 9994: 9928: 9885: 9875: 9855: 9529: 9438: 9433: 9427: 9419: 9380: 8735: 8519: 8135: 7970: 7775: 7746: 7685: 7672:(1936). "Die Lehre von Geraden und Ungeraden im neunten Buch der euklidischen Elemente". 7605: 6429: 6296:, pp. 42–50. A slightly more explicit description of the kuṭṭaka was later given in 6069: 5505: 5427:, viz., seeking correspondents, visiting foreign colleagues, building private libraries ( 4675: 4665: 4633: 4340: 4104: 3599: 3536: 3527:—are relatively well understood. Their classification was the object of the programme of 3073: 2821: 2615: 2466: 1512: 1281: 1081: 996: 866: 838:(which may be identified with real numbers, whether rational or not), on the other hand. 824:, who was expelled or split from the Pythagorean sect. This forced a distinction between 817: 692: 306: 294: 9619: 7465: 7388: 7365:

Schumayer, Daniel; Hutchinson, David A. W. (2011). "Physics of the Riemann Hypothesis".

5492:

This is the case for small sieves (in particular, some combinatorial sieves such as the

2860:) also occupies an increasingly central place in the toolbox of analytic number theory. 2583:

The development of much of modern mathematics necessary for basic modern number theory:

11256: 11155: 11031: 10988: 10897: 10839: 10824: 10814: 10606: 10405: 10289: 10208: 10148: 10080: 10070: 10009: 9984: 9860: 9817: 9812: 9575: 9519: 9423: 9340: 9133: 8968: 8867: 8832: 8814: 8646: 8499: 8477: 8436: 8418: 7941: 7813: 7748:

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara

7550: 7400: 7374: 7345: 5318:. This notation is actually much later than Fermat's; it first appears in section 1 of 5004: 4935: 4905: 4901: 4752: 4721: 4701: 4669: 4647: 4607: 4574: 4477: 4457: 4404: 4347:, however, is a contemporary term for much the same domain as that covered by the term 4297: 4277: 4257: 4173: 4084: 3921: 3528: 3365: 3345: 3226: 3160: 3140: 3120: 3077: 2951: 2837: 2780: 2205: 2190: 2025: 1451: 1411: 1292:

While Greek astronomy probably influenced Indian learning, to the point of introducing

1120: 936: 732: 531: 436: 366: 235: 139: 8863:

lacks Truesdell's intro, which is reprinted (slightly abridged) in the following book:

7573:

Dauben, Joseph W. (2007), "Chapter 3: Chinese Mathematics", in Katz, Victor J. (ed.),

5558: 2809:

Some subjects generally considered to be part of analytic number theory, for example,

2537: 1456: 11276: 11206: 11185: 11147: 10955: 10922: 10902: 10601: 10513: 10387: 10284: 10004: 9989: 9933: 9880: 9702: 9489: 9257: 9247: 9243: 9221: 9178: 9174: 9147: 9088: 9020: 8914: 8883: 8848: 8739: 8729: 8725: 8650: 8593: 8485: 8451: 8404: 8373: 8320: 8297: 8268: 8264:

Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China

8215: 8168: 8139: 8113: 8085: 8028: 7982: 7927: 7903: 7889: 7872: 7836: 7779: 7703: 7699: 7693: 7634: 7609: 7578: 7542: 7481: 7433: 7404: 7322: 7312: 7274: 7244: 7219: 7179: 7155: 6501: 5993: 5501: 5423:, pp. 159, 161). (There were already some recognisable features of professional 4998: 4955: 4832:, or, what is the same, for finding the quantities whose existence is assured by the 4780: 3580: 3547: 3532: 3178: 2857: 2687: 2381:)—defining their equivalence relation, showing how to put them in reduced form, etc. 2284: 1420: 862: 858: 785: 314: 231: 8790: 8787: 8784: 8781: 8774: 7714: 7554: 7419: 7260: 717:, presumably for actual use as a "table", for example, with a view to applications. 11109: 10993: 10960: 10755: 10684: 10573: 10559: 10554: 10503: 10490: 10415: 10368: 10218: 10193: 10065: 9913: 9850: 9494: 9479: 9207: 9128: 8960: 8766: 8677: 8638: 8344: 8002: 7884: 7850: 7805: 7763: 7532: 7524: 7473: 7425: 7392: 7266: 7203: 7189: 5067: 4921: 4874:

which can solve all Diophantine equations. In particular, this means that, given a

4761: 2864: 2799: 2776: 2702:

proof may be longer and more difficult for most readers than a non-elementary one.

2695: 2671: 2666: 2632: 2611: 2584: 2213: 2125:(1770), soon improved by Euler himself); the lack of non-zero integer solutions to 1664: 1577: 1558: 1552: 1475:] represents knowledge through reason and Galileo knowledge through the senses. 846: 842: 759: 376: 371: 325: 286: 7854: 4911:

Number theory has now several modern applications spanning diverse areas such as:

4664:

being studied need not be sets of integers, but rather subsets of non-commutative

1168:

survive in the original Greek and four more survive in an Arabic translation. The

883: 11180: 11074: 11046: 10940: 10892: 10877: 10862: 10717: 10712: 10659: 10549: 10523: 10475: 10420: 10158: 10085: 10014: 9807: 9508: 9484: 9399: 9231: 9211: 9168: 9137: 9078: 9014: 8904: 8871: 8844: 8836: 8367: 8310: 8262: 8233: 8145: 8123: 7974: 7921: 7832: 7826: 7724:

The Āryabhaṭīya of Āryabhaṭa: An ancient Indian work on Mathematics and Astronomy

7656: 7595: 7575:

The Mathematics of Egypt, Mesopotamia, China, India and Islam : A Sourcebook

7477: 6495: 6263:

Any early contact between Babylonian and Indian mathematics remains conjectural (

4888: 4860: 4322:: if a number can be approximated better than any algebraic number, then it is a 3576: 2767: 2562: 2003:; this claim appears in his annotations in the margins of his copy of Diophantus. 887: 310: 157: 7004: 6456:, p. 118. This was more so in number theory than in other areas (remark in 3459:

Number fields are often studied as extensions of smaller number fields: a field

2598:

The rough subdivision of number theory into its modern subfields—in particular,

1123:). As far as it is known, such equations were first successfully treated by the 11286: 11190: 11089: 10935: 10907: 10236: 10163: 9870: 9687: 9545: 9464: 9348: 9277: 9217: 8658: 8207: 8010: 7591: 7512: 7159: 5582: 4871: 4849: 4629: 3865: 3543: 2910: 2833: 2618:

theory; see below. A conventional starting point for analytic number theory is

2606:

Algebraic number theory may be said to start with the study of reciprocity and

2525: 2494: 2332: 2021: 1569: 1508: 1496: 1058: 878: 789: 185: 8779:. (4 Vols.) (in French and Latin). Paris: Imprimerie Gauthier-Villars et Fils. 7528: 7396: 7326: 2705: 1428: 914:

which was translated into English in early 19th century by British missionary

830:(integers and the rationals—the subjects of arithmetic), on the one hand, and 320:

Integers can be considered either in themselves or as solutions to equations (

11354: 11175: 10470: 9956: 9908: 9504: 9356: 9129: 8696: 8563: 8543: 8503: 8455: 8432: 7546: 7163: 5436: 5022: 4369: 3524: 2841: 2795: 2772: 2748: 2694:) are often seen as quite enlightening but not elementary, in spite of using 2607: 2592: 1465: 850: 340:

in relation to rational numbers; for example, as approximated by the latter (

324:). Questions in number theory are often best understood through the study of 9010: 8237: 7470:

Applications of Algebraic Geometry to Coding Theory, Physics and Computation

5967:, pp. 87–90) sustains the view that Thales knew Babylonian mathematics. 4571:, say? Should it be possible to write large integers as sums of elements of 3587:-dimensional space. In Diophantine geometry, one asks whether there are any 2710: 2679: 11271: 10930: 9966: 9961: 9865: 9550: 9474: 9374: 8762: 8681: 8258: 7689: 7669: 7306: 7218:, Stefan Andrus Burr, George E. Andrews, American Mathematical Soc., 1992, 6609: 5522: 5105: 5017: 4983: 4958:", which is the basis for most modern Western music, involves dividing the 4948: 4915: 4904:; in other words, number theory also has applications to the continuous in 4892: 4856: 4765: 4757: 4637: 4452: 3690:

are both rational. This is the same as asking for all integer solutions to

3504: 3449: 3061: 2872: 2853: 2845: 2810: 2794:

in terms of its tools, as the study of the integers by means of tools from

2683: 2644: 2588: 2556: 2296: 2217: 1480: 1293: 1112: 854: 428: 302: 51: 9723: 7979:

The Shaping of Arithmetic after C.F. Gauss's "Disquisitiones Arithmeticae"

7831:. Graduate Texts in Mathematics. Vol. 50 (reprint of 1977 ed.). 7429: 7270: 3053:(say) is an algebraic number. Fields of algebraic numbers are also called 739:

learned mathematics from the Babylonians. Much earlier sources state that

11261: 10887: 10799: 10168: 9832: 9755: 9074: 9044: 8583: 8363: 8099: 8014: 6297: 5497: 5075: 5046: 3184:

The grounds of the subject were set in the late nineteenth century, when

2868: 2849: 2714: 2623: 2517: 1564:

Over his lifetime, Fermat made the following contributions to the field:

1528: 1164: 905: 337: 148: 74: 55: 7946:"Elementary Proof of the Prime Number Theorem: a Historical Perspective" 7828:

Fermat's Last Theorem: a Genetic Introduction to Algebraic Number Theory

7674:

Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik

5431:, pp. 160–161). Matters started to shift in the late 17th century ( 731:

was exceptionally well developed and included the foundations of modern

11281: 11211: 10804: 10544: 10400: 10153: 10032: 9827: 9644: 9555: 9514: 9366: 9016:

Number Theory: an Approach Through History – from Hammurapi to Legendre

8972: 8796: 8642: 7817: 7537: 6585: 6573: 6534:, Vol. II, p. 209, Letter XLVI from Fermat to Frenicle, 1640, cited in 5936: 5561:; in modern language, the main outcome of his work is that an equation 5493: 5202:

This is the last problem in Sunzi's otherwise matter-of-fact treatise.

4977: 3542:

An example of an active area of research in algebraic number theory is

2686:. The term is somewhat ambiguous: for example, proofs based on complex 2500: 1460: 1423:, or "cyclic method") for solving Pell's equation was finally found by 1415: 1127:. It is not known whether Archimedes himself had a method of solution. 1108: 959: 755: 736: 357: 277: 153: 143: 62:

between being prime and being a value of certain quadratic polynomials.

40: 7774:. Graduate Texts in Mathematics. Vol. 74 (revised 3rd ed.). 4425:

At times, a non-rigorous, probabilistic approach leads to a number of

3650:

one would like to know its rational solutions; that is, its solutions

2816:

The following are examples of problems in analytic number theory: the

10786: 10747: 8663:"Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" 8567: 5529:

consists of two elements: the identity element (taking every element

5112:, p. 202) with a view to "perhaps knocking off its pedestal" ( 4426: 2840:. Some of the most important tools of analytic number theory are the 2759:): dark colors denote values close to zero and hue gives the value's 2595:—accompanied by greater rigor in analysis and abstraction in algebra. 2316: 1524: 1504: 1299:Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences 33:

Number Theory: An Approach Through History from Hammurapi to Legendre

8964: 7809: 5399: 5116:, p. 167); at the same time, it settles to the conclusion that 4855:

The difficulty of a computation can be useful: modern protocols for

924: 894:.) The result was later generalized with a complete solution called 58:

serves to illustrate it, hinting, in particular, at the conditional

10847: 10337: 10057: 9976: 9903: 9203: 2888: 1548: 1135: 989:

have come to mean.) It is through one of Plato's dialogues—namely,

955: 125: 115: 46: 8629:

Rashed, Roshdi (1980). "Ibn al-Haytham et le théorème de Wilson".

7379: 6906: 5435:, p. 161); scientific academies were founded in England (the 4401:

uses the fact that whatever happens with probability greater than

2291: 1807:

Fermat stated and proved (by infinite descent) in the appendix to

792:

gave great importance to the odd and the even. The discovery that

538:

of the diagonal which has been subtracted such that the width..."

530:. The triples are too many and too large to have been obtained by 9842: 8809:. London: J.M. Watkins. Archived from the original on 2011-07-21. 8523: 8394: 6851: 4931: 2320:

Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition

2012: 1581: 1104: 290: 120: 9291: 9115:(reprint of the 1954 ed.). Mineola, NY: Dover Publications. 8301: 7977:. In Goldstein, C.; Schappacher, N.; Schwermer, Joachim (eds.). 6967: 5541:

to itself) and complex conjugation (the map taking each element

5394:); this fact (which, in modern language, makes the residues mod 9454: 9286: 8141:

A History of Greek Mathematics, Volume 1: From Thales to Euclid

7513:"Aesthetics, Dynamics, and Musical Scales: A Golden Connection" 6460:, p. 284). Bachet's own proofs were "ludicrously clumsy" ( 4959: 2736: 2216:, most of his work is restricted to the formal manipulation of 826: 740: 419: 313:), or defined as generalizations of the integers (for example, 7510: 6734:, p. 174. Euler was generous in giving credit to others ( 6035: 2698:, rather than complex analysis as such. Here as elsewhere, an 2552: 1523:

Other than a treatise on squares in arithmetic progression by

9083:(rev. by D.R. Heath-Brown and J.H. Silverman, 6th ed.). 8229: 6112:, pp. 219–220, which contains a full translation of the 5918: 5877: 5729: 5375: 5319: 5263: 3869: 3860: 2102: 1631: 1374: 1324: 1148: 941: 744: 432: 9053:

Creative Commons Attribution-ShareAlike 3.0 Unported License

8878:. Volume 2 of MAA tercentenary Euler celebration. New York: 7016: 6992: 6801: 6765: 5398:

into a group, and which was already known to Āryabhaṭa; see

2536:

In this way, Gauss arguably made a first foray towards both

351:. By the early twentieth century, it had been superseded by 6554:, p. 63. All of the following citations from Fermat's 6288:Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in: 6163: 4359:) rather than to techniques in Diophantine approximations. 3938:

is a polynomial in two variables) depends crucially on the

2852:(or, rather, the study of their properties). The theory of 2512:(1798), Carl Friedrich Gauss (1777–1855) proved the law of 1042:{\displaystyle {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}} 9216:. Pure and Applied Mathematics. Vol. 20. Boston, MA: 9069:

Two of the most popular introductions to the subject are:

8550:(1968). "The Fragments of the Works of Ya'qub ibn Tariq". 7206:, editor, pp. 269–378, American Mathematical Society 1996. 6789: 5984:. On Thales, see Eudemus ap. Proclus, 65.7, (for example, 3743:; any solution to the latter equation gives us a solution 3531:, which was initiated in the late 19th century (partly by 2039:(generalised by Euler to non-prime moduli); the fact that 981:

he meant, in part, theorising on number, rather than what

939:. In the case of number theory, this means, by and large, 7150: 6407: 6247: 6245: 1471: 816:

is irrational is credited to the early Pythagoreans (pre-

6657: 6218: 2948:

with rational coefficients; for example, every solution

7168:

Mathematical Developments Arising from Hilbert Problems

6957: 6955: 6894: 6882: 6870: 6813: 6777: 6713: 6633: 4604:(which concerns itself with certain very specific sets 4327: 2547: 1441: 8498: 8312:

The Mathematical Career of Pierre de Fermat, 1601–1665

6830: 6828: 6645: 6351: 6339: 6327: 6315: 6270: 6242: 5763: 5402:) was familiar to Fermat thanks to its rediscovery by 2610:, but truly came into its own with the development of 9469: 9459: 9073: 8876:

The Genius of Euler: reflections on his life and work

8161:

Religion, Learning and Science in the 'Abbasid Period

7472:, Dordrecht: Springer Netherlands, pp. 255–270, 7132: 6435: 6230: 6075: 5906: 5792: 5786:, p. 189. Other sources give the modern formula 5654: 5600: 5361: 5252: 5217:(ca. 100 CE), who wrote a primitive but influential " 4800: 4724: 4704: 4678: 4650: 4610: 4577: 4500: 4480: 4460: 4407: 4339:

Diophantine geometry should not be confused with the

4300: 4280: 4260: 4196: 4176: 4135: 4107: 4087: 4039: 3993: 3952: 3924: 3883: 3817: 3783: 3749: 3696: 3656: 3607: 3421: 3388: 3368: 3348: 3281: 3249: 3229: 3202: 3163: 3143: 3123: 3090: 2974: 2954: 2919: 2469: 2401: 2341: 2236: 2131: 2091: 2045: 1983: 1930: 1870: 1817: 1760: 1713: 1673: 1607: 1355: 1305: 1226: 1178: 1124: 1005: 891: 798: 765: 754:, propositions 21–34 are very probably influenced by 695: 553: 483: 445: 7981:. Berlin & Heidelberg: Springer. pp. 3–66. 7969: 7796:(November 1983). "Euler and Quadratic Reciprocity". 7041: 6952: 6942: 6929: 6912: 5569:) = 0 can be solved by radicals (that is, 4994: 4598:. This is a presently coalescing field; it subsumes 2576:

The rise to self-consciousness of number theory (or

1518: 9202: 8570:(1970). "The Fragments of the Works of al-Fazari". 7762: 7421:

Error-Correcting Codes: A Mathematical Introduction

7364: 7034:See the comment on the importance of modularity in 6986: 6973: 6825: 6741: 6669: 6621: 6087: 3591:(points all of whose coordinates are rationals) or 2631:analysis in number theory comes later: the work of 1287: 1284:to which rational or integer solutions are sought. 974:("They say Plato learned all things Pythagorean"). 54:is a central point of study in number theory. This 7466:"Communication Networks and Hilbert Modular Forms" 7262:An Introduction to Number Theory with Cryptography 7120: 6597: 6023: 5862: 5716: 5632: 5500:; the study of the latter now includes ideas from 5386: 5326:. Fermat's little theorem is a consequence of the 5274: 4824: 4730: 4710: 4690: 4656: 4616: 4583: 4563: 4486: 4466: 4413: 4306: 4286: 4274:is large. This question is of special interest if 4266: 4246: 4182: 4162: 4121: 4093: 4066: 4020: 3979: 3946:can be defined as follows: allow the variables in 3930: 3910: 3849: 3803: 3769: 3735: 3674: 3642: 3440: 3407: 3374: 3354: 3335:{\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})} 3334: 3267: 3235: 3215: 3169: 3149: 3129: 3109: 3045: 2960: 2940: 2481: 2455: 2373: 2265: 2170: 2113: 2077: 1995: 1969: 1909: 1856: 1795: 1739: 1699: 1645: 1391: 1341: 1266: 1212: 1057:, and was thus arguably a pioneer in the study of 1041: 841:The Pythagorean tradition spoke also of so-called 808: 775: 709: 674: 522: 469: 8709:. London: Kegan, Paul, Trench, Trübner & Co. 8437:"Mathematics in India: reviewed by David Mumford" 8369:Multiplicative Number Theory: I, Classical Theory 8358: 8267:(revised ed.). Singapore: World Scientific. 7114: 5481: 4474:contain many elements in arithmetic progression: 4074:. Other geometrical notions are just as crucial. 925:Classical Greece and the early Hellenistic period 11352: 9166: 8819:: CS1 maint: bot: original URL status unknown ( 7916: 7577:, Princeton University Press, pp. 187–384, 5745: 4844:("pulveriser"), without a proof of correctness. 4136: 2311: 2186:of which Euler also proved by a related method). 1115:. The epigram proposed what has become known as 1088:, Prop. VII.2) and the first known proof of the 534:. The heading over the first column reads: "The 9198:Popular choices for a second textbook include: 8993: 8206: 7216:The Unreasonable Effectiveness of Number Theory 7089: 7060: 7035: 6998: 6041: 6017: 5964: 5948: 5924: 5883: 5871: 5477: 5222: 5084:(1938): "We proposed at one time to change to 3857:(a circle of radius 1 centered on the origin). 2913:that is a solution to some polynomial equation 8734:. Graduate Texts in Mathematics. Vol. 7. 8163:. The Cambridge history of Arabic literature. 6494:Faulkner, Nicholas; Hosch, William L. (2017). 6108:, Chapter 3, Problem 26. This can be found in 4741: 4379: 2620:Dirichlet's theorem on arithmetic progressions 2393:Dirichlet's theorem on arithmetic progressions 1069:as being largely based on Theaetetus's work.) 10377:Note: This template roughly follows the 2012 10353: 9739: 9660: 9307: 8484:. Vol. 9. New York: Dover Publications. 7859:(in French and Latin). Toulouse: Joannis Pech 7684: 6493: 6169: 5863:{\displaystyle (p^{2}-q^{2},2pq,p^{2}+q^{2})} 4790:) it appears as Proposition 2 of Book VII in 4247:{\displaystyle |x-a/q|<{\frac {1}{q^{c}}}} 2670:generally denotes a method that does not use 1754:In 1657, Fermat posed the problem of solving 1538: 899: 253: 9043:This article incorporates material from the 8906:Euler Through Time: A New Look at Old Themes 8827:Iamblichus#List of editions and translations 8761: 8695: 8610:Suanjing shi shu (Ten Mathematical Classics) 8526:. The Internet Encyclopaedia of Philosophy. 8444:Notices of the American Mathematical Society 8423:: CS1 maint: multiple names: authors list ( 7900:Classics in the History of Greek Mathematics 7265:(2nd ed.). Chapman and Hall/CRC. 2018. 6615: 6591: 6579: 6547: 6531: 6477: 6377: 4951:, which is a part of analytic number theory. 4840:(5th–6th century CE) as an algorithm called 3046:{\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0} 1463:as seen by the West: on the frontispiece of 1392:{\displaystyle n\equiv a_{2}{\bmod {m}}_{2}} 1342:{\displaystyle n\equiv a_{1}{\bmod {m}}_{1}} 9753: 8902: 8518: 8400:A Commentary on Book 1 of Euclid's Elements 8098: 7920:; Waterhouse, William C. (trans.) (1966) . 7172:Proceedings of Symposia in Pure Mathematics 7010: 6807: 6771: 6759: 6735: 6703: 6691: 6663: 5989: 5581:) = 0 has a Galois group that is 5464: 5452: 5108:. Robson's article is written polemically ( 5089: 4564:{\displaystyle a+b,a+2b,a+3b,\ldots ,a+10b} 4436: 2659: 2395:. He gave a full treatment of the equation 2198:First steps towards analytic number theory. 1646:{\displaystyle a^{p-1}\equiv 1{\bmod {p}}.} 10360: 10346: 9746: 9732: 9667: 9653: 9609: 9599: 9314: 9300: 9107: 8843:(reprint of 1840 5th ed.). New York: 8510:. American Oriental Series. Vol. 29. 8476: 7741: 7350:: CS1 maint: location missing publisher ( 6369: 6357: 6305: 5900: 5896: 5104:, p. 201. This is controversial. See 2894: 1924:) to have shown there are no solutions to 1580:; these topics led him to work on integer 972:Platonem ferunt didicisse Pythagorea omnia 857:, etc., are seen now as more natural than 365:"; it has also acquired other meanings in 260: 246: 8954: 8866: 8831: 8001: 7897: 7888: 7628: 7536: 7378: 7101: 7072: 7059:See, for example, the initial comment in 7022: 6158: 6081: 6062: 5992:, p. 1. Proclus was using a work by 5970: 5460: 4954:Study of musical scales: the concept of " 4372:began only in the 1930s and 1940s, while 4077:There is also the closely linked area of 2725: 2717:in 1985, when Erdős was 72 and Tao was 10 397:is commonly preferred as an adjective to 9674: 9139:An introduction to the theory of numbers 9080:An introduction to the theory of numbers 8290:Elementary Introduction to Number Theory 8105:An Introduction to the Theory of Numbers 8084:. Grand Rapids, Michigan: Phanes Press. 7940: 7092:, p. 1: "However much stronger...". 7047: 5930: 5455:, p. 7). In this context, the term 5081:An Introduction to the Theory of Numbers 5078:and Wright wrote in the introduction to 4751: 3859: 3539:) and carried out largely in 1900–1950. 2766: 2735: 2704: 2561: 2551: 2499: 2315: 2290: 2011: 1547: 1487:, which may or may not be Brahmagupta's 1455: 1134: 418: 309:constructed from integers (for example, 45: 9121: 8613:(in Chinese). Beijing: Zhonghua shuju. 8582: 8562: 8542: 8431: 8308: 8181: 8158: 8076: 8061:. Alpine, New Jersey: Platonist Press. 8047: 7870: 7824: 7792: 7646: 7590: 7304: 7138: 7088:See the remarks in the introduction to 6961: 6795: 6762:, pp. 45–55; see also chapter III. 6725: 6481: 6473: 6457: 6401: 6397: 6393: 6387: 6376:, p. 302. See also the preface in 6373: 6345: 6333: 6321: 6289: 6276: 6264: 6257: 6251: 6004:, p. xxx on Proclus's reliability. 5981: 5960: 5944: 5912: 5717:{\displaystyle (a+bi)^{2}=(c+di)^{3}+7} 5314:leave the same residue when divided by 3553: 2287:, whose study he had himself initiated. 1479:In the early ninth century, the caliph 743:and Pythagoras traveled and studied in 361:is used by the general public to mean " 14: 11353: 11070:Knowledge representation and reasoning 8806:Life of Pythagoras or, Pythagoric Life 8795: 8657: 8628: 8392: 8257: 8228: 8082:The Pythagorean Sourcebook and Library 8020:The Princeton Companion to Mathematics 7849: 7721:Clark, Walter Eugene (trans.) (1930). 7668: 7597:Introduction to analytic number theory 7572: 7417: 6567: 6413: 6146: 6134: 6127: 6121: 6109: 6058: 6029: 6013: 6007: 6001: 5985: 5783: 5771: 5181: 5147: 5130: 5121: 5113: 5109: 5101: 4594:These questions are characteristic of 4429:algorithms and open problems, notably 4336:have been shown to be transcendental. 2456:{\displaystyle ax^{2}+by^{2}+cz^{2}=0} 1399:could be solved by a method he called 911:Mathematical Treatise in Nine Sections 735:. Late Neoplatonic sources state that 289:devoted primarily to the study of the 11095:Philosophy of artificial intelligence 10341: 9727: 9648: 9295: 8933: 8752: 8724: 8631:Archive for History of Exact Sciences 8393:Morrow, Glenn Raymond (trans., ed.); 8339: 8134: 7720: 7463: 7200:The Collected Works of Julia Robinson 7126: 6428:, 1621, following a first attempt by 6381: 6309: 6293: 6224: 6093: 5387:{\displaystyle xa\equiv 1{\bmod {p}}} 5334:of an element of a group divides the 5063: 5050: 4779:goes back only to certain readers of 2863:One may ask analytic questions about 2779:. The region in grey is the standard 2200:In his work of sums of four squares, 414: 10421:Energy consumption (Green computing) 10367: 10308: 9142:(reprint of the 5th 1991 ed.). 9009: 8606: 8287: 8194:from the original on 2 December 2013 8188:Stanford Encyclopaedia of Philosophy 6946: 6900: 6888: 6876: 6834: 6819: 6783: 6747: 6738:, p. 14), not always correctly. 6731: 6719: 6707: 6687: 6675: 6651: 6639: 6627: 6603: 6559: 6551: 6535: 6461: 6453: 6441: 6236: 6117: 6047: 5751: 5448: 5432: 5428: 5420: 5407: 5275:{\displaystyle a\equiv b{\bmod {m}}} 3479:. (For example, the complex numbers 3084:consists of all numbers of the form 2654: 2548:Maturity and division into subfields 2114:{\displaystyle p\equiv 1{\bmod {4}}} 1568:One of Fermat's first interests was 1442:Arithmetic in the Islamic golden age 347:The older term for number theory is 11100:Distributed artificial intelligence 10379:ACM Computing Classification System 10320: 9708:Mersenne primes and perfect numbers 8880:Mathematical Association of America 8839:. In Hewlett, John (trans.) (ed.). 8530:from the original on 6 January 2016 7676:. Abteilung B:Studien (in German). 6941:See the discussion in section 5 of 6840: 5234:Here, as usual, given two integers 5070:still had to specify that he meant 5028:List of number theoretic algorithms 4836:) it first appears in the works of 2883:, which are generalizations of the 2182:of Fermat's last theorem, the case 1804:could be found by infinite descent. 435:, ca. 1800 BC) contains a list of " 24: 10612:Integrated development environment 9064: 8997:; Dresden, Arnold (trans) (1961). 8841:Leonard Euler, Elements of Algebra 8182:Huffman, Carl A. (8 August 2011). 7975:"A book in search of a discipline" 7602:Undergraduate Texts in Mathematics 7107: 6858:from the original on 17 March 2016 5976:Herodotus (II. 81) and Isocrates ( 4362: 3491:are an extension of the rationals 2335:in full generality (as opposed to 1431:'s Bīja-gaṇita (twelfth century). 1139:Title page of the 1621 edition of 25: 11372: 11080:Automated planning and scheduling 10617:Software configuration management 9321: 9271: 8872:"Leonard Euler, Supreme Geometer" 8837:"Leonard Euler, Supreme Geometer" 7649:Mathematical Reviews (MathSciNet) 5517:The Galois group of an extension 3736:{\displaystyle a^{2}+b^{2}=c^{2}} 2639:is the canonical starting point; 2171:{\displaystyle x^{4}+y^{4}=z^{2}} 1970:{\displaystyle x^{n}+y^{n}=z^{n}} 1910:{\displaystyle x^{3}+y^{3}=z^{3}} 1857:{\displaystyle x^{4}+y^{4}=z^{4}} 1519:Western Europe in the Middle Ages 1153:Claude Gaspard Bachet de Méziriac 523:{\displaystyle a^{2}+b^{2}=c^{2}} 27:Mathematics of integer properties 11334: 11324: 11315: 11314: 10319: 10307: 10296: 10295: 10283: 9628: 9618: 9608: 9598: 9589: 9588: 7504: 7457: 7411: 7358: 7298: 7253: 7237: 7228: 7209: 6943:Goldstein & Schappacher 2007 6930:Goldstein & Schappacher 2007 6928:; the translation is taken from 6913:Goldstein & Schappacher 2007 6848:"Andrew Wiles on Solving Fermat" 6566:originally prepared by his son ( 6480:, Vol. II, pp. 72, 74, cited in 4997: 4644:is also used; however, the sets 1511:knew what would later be called 1499:(820–912). Part of the treatise 1495:, was translated into Arabic by 1288:Āryabhaṭa, Brahmagupta, Bhāskara 869:(17th to early 19th centuries). 230: 30:For the book by André Weil, see 11325: 10728:Computational complexity theory 10204:Computational complexity theory 9713:Gaussian integer factorizations 9051:", which is licensed under the 8982:from the original on 2012-07-15 8757:. New York: Dover Publications. 8713:from the original on 2016-03-03 8617:from the original on 2013-11-02 8572:Journal of Near Eastern Studies 8552:Journal of Near Eastern Studies 8482:The Exact Sciences in Antiquity 8465:from the original on 2021-05-06 8292:(2nd ed.). Lexington, VA: 8246:from the original on 2011-07-09 8065:from the original on 2020-02-29 7973:; Schappacher, Norbert (2007). 7958:from the original on 2016-03-03 7494:from the original on 2023-03-01 7446:from the original on 2023-03-01 7333:from the original on 2023-03-01 7287:from the original on 2023-03-01 7144: 7095: 7082: 7066: 7053: 7028: 6987:Davenport & Montgomery 2000 6979: 6974:Davenport & Montgomery 2000 6935: 6918: 6753: 6697: 6681: 6541: 6525: 6514:from the original on 2023-03-01 6487: 6467: 6447: 6419: 6363: 6282: 6209: 6200: 6189:from the original on 2016-12-11 6175: 6152: 6140: 6099: 5588: 5511: 5486: 5470: 5413: 5228: 5206: 5170: 5137: 4882: 4374:computational complexity theory 3583:, or some other such object in 2622:(1837), whose proof introduced 2033:Proofs for Fermat's statements. 1491:). Diophantus's main work, the 285:in older usage) is a branch of 10519:Network performance evaluation 9367:analytic theory of L-functions 9345:non-abelian class field theory 9001:. Vol. 1 or 2. New York: 8592:. Princeton University Press. 8403:. Princeton University Press. 8372:. Cambridge University Press. 7871:Friberg, Jöran (August 1981). 6120:). See also the discussion in 5889: 5857: 5793: 5777: 5757: 5699: 5683: 5671: 5655: 5594:If we want to study the curve 5095: 5056: 5039: 4672:, in which case the growth of 4326:. It is by this argument that 4220: 4198: 4151: 4139: 4055: 4043: 4009: 3997: 3968: 3956: 3899: 3887: 3669: 3657: 3643:{\displaystyle x^{2}+y^{2}=1,} 3483:are an extension of the reals 3441:{\displaystyle 1-{\sqrt {-5}}} 3408:{\displaystyle 1+{\sqrt {-5}}} 3329: 3310: 3307: 3288: 3110:{\displaystyle a+b{\sqrt {d}}} 3002: 2988: 2929: 2923: 2879:by means of an examination of 2567:Peter Gustav Lejeune Dirichlet 2491:Peter Gustav Lejeune Dirichlet 1796:{\displaystyle x^{2}-Ny^{2}=1} 1267:{\displaystyle f(x,y,z)=w^{2}} 1248: 1230: 1194: 1182: 958:, PE X, chapter 4 mentions of 464: 446: 13: 1: 10883:Multimedia information system 10868:Geographic information system 10858:Enterprise information system 10454:Computer systems organization 9254:Graduate Texts in Mathematics 8943:American Mathematical Monthly 8911:American Mathematical Society 8859:This Google books preview of 8755:History of Mathematics, Vol I 8210:; Kowalski, Emmanuel (2004). 8186:. In Zalta, Edward N. (ed.). 7902:. Berlin: Kluwer (Springer). 7653:American Mathematical Society 7517:Journal of New Music Research 7176:American Mathematical Society 7115:Montgomery & Vaughan 2007 6400:, pp. 103–123, cited in 6292:, pp. 134–140. See also 5738: 5633:{\displaystyle y^{2}=x^{3}+7} 5482:Montgomery & Vaughan 2007 5086:An introduction to arithmetic 4973:American Mathematical Society 4640:, and other fields. The term 3850:{\displaystyle x^{2}+y^{2}=1} 2493:, and crediting both him and 2374:{\displaystyle mX^{2}+nY^{2}} 2312:Lagrange, Legendre, and Gauss 2078:{\displaystyle p=x^{2}+y^{2}} 1448:Mathematics in medieval Islam 1130: 685:which is implicit in routine 305:as well as the properties of 11242:Computational social science 10830:Theoretical computer science 10650:Software development process 10426:Electronic design automation 10411:Very Large Scale Integration 9391:Transcendental number theory 8936:"Archimedes' Cattle Problem" 8874:. In Dunham, William (ed.). 8508:Mathematical Cuneiform Texts 7890:10.1016/0315-0860(81)90069-0 7825:Edwards, Harold M. (2000) . 7772:Multiplicative Number Theory 7478:10.1007/978-94-010-1011-5_13 7013:, sections 2.5, 3.1 and 6.1. 6550:, Vol. II, p. 204, cited in 6500:. Encyclopaedia Britannica. 5521:consists of the operations ( 4320:transcendental number theory 3987:to be complex numbers; then 3216:{\displaystyle {\sqrt {-5}}} 3072:can be restated in terms of 2830:Hardy–Littlewood conjectures 2641:Jacobi's four-square theorem 2602:and algebraic number theory. 2266:{\displaystyle x^{2}+Ny^{2}} 1597:is not divisible by a prime 1213:{\displaystyle f(x,y)=z^{2}} 7: 11065:Natural language processing 10853:Information storage systems 9614:List of recreational topics 9386:Computational number theory 9371:probabilistic number theory 9282:Encyclopedia of Mathematics 8701:Bīrūni, ̄Muḥammad ibn Aḥmad 8607:Qian, Baocong, ed. (1963). 7923:Disquisitiones Arithmeticae 7729:University of Chicago Press 7090:Iwaniec & Kowalski 2004 7061:Iwaniec & Kowalski 2004 7036:Iwaniec & Kowalski 2004 6999:Iwaniec & Kowalski 2004 6926:Disquisitiones Arithmeticae 5764:Neugebauer & Sachs 1945 5478:Iwaniec & Kowalski 2004 5324:Disquisitiones Arithmeticae 4990: 4978:Cole Prize in Number Theory 4748:Computational number theory 4742:Computational number theory 4399:probabilistic combinatorics 4386:Probabilistic number theory 4380:Probabilistic number theory 4294:is an algebraic number. If 4163:{\displaystyle \gcd(a,q)=1} 3872:in four-dimensional space.) 3070:Disquisitiones arithmeticae 2614:and early ideal theory and 2524:established a link between 2510:Disquisitiones Arithmeticae 1740:{\displaystyle a^{2}+b^{2}} 1700:{\displaystyle a^{2}+b^{2}} 1158:Very little is known about 1117:Archimedes's cattle problem 1072:Euclid devoted part of his 809:{\displaystyle {\sqrt {2}}} 776:{\displaystyle {\sqrt {2}}} 10: 11377: 10981:Human–computer interaction 10951:Intrusion detection system 10863:Social information systems 10848:Database management system 10254:Films about mathematicians 8995:van der Waerden, Bartel L. 8934:Vardi, Ilan (April 1998). 8903:Varadarajan, V.S. (2006). 8506:; Götze, Albrecht (1945). 8317:Princeton University Press 8165:Cambridge University Press 8025:Princeton University Press 7698:(2nd ed.). New York: 7633:, Dover Publications Inc, 7629:Libbrecht, Ulrich (1973), 7565: 6057:, p. 147 B, (for example, 5219:Introduction to Arithmetic 4745: 4440: 4397:It is sometimes said that 4383: 4170:) a good approximation to 4079:Diophantine approximations 3557: 3268:{\displaystyle 6=2\cdot 3} 3243:can be factorised both as 2898: 2729: 1809:Observations on Diophantus 1749:method of infinite descent 1539:Early modern number theory 1445: 928: 876:appears as an exercise in 409: 404: 328:objects (for example, the 170:Relationship with sciences 38: 29: 11310: 11247:Computational engineering 11222:Computational mathematics 11199: 11146: 11108: 11055: 11017: 10979: 10921: 10838: 10784: 10746: 10698: 10635: 10568: 10532: 10489: 10453: 10386: 10375: 10277: 10227: 10184: 10094: 10056: 10023: 9975: 9947: 9894: 9841: 9823:Philosophy of mathematics 9798: 9763: 9683: 9584: 9566:Diophantine approximation 9538: 9525:Chinese remainder theorem 9447: 9329: 9256:. Vol. 7. Springer. 9167:Kenneth H. Rosen (2010). 9113:Elements of Number Theory 8512:American Oriental Society 8345:"Algebraic Number Theory" 8315:(Reprint, 2nd ed.). 8234:Jowett, Benjamin (trans.) 7529:10.1076/jnmr.31.1.51.8099 7397:10.1103/RevModPhys.83.307 7367:Reviews of Modern Physics 6206:Metaphysics, 1.6.1 (987a) 6170:Boyer & Merzbach 1991 6016:, p. 533, cited in: 6000:. See also introduction, 5347:not divisible by a prime 4966: 4834:Chinese remainder theorem 3157:are rational numbers and 2273:, some of it prefiguring 1572:(which appear in Euclid, 1543: 931:Ancient Greek mathematics 900: 874:Chinese remainder theorem 382:floating-point arithmetic 342:Diophantine approximation 11257:Computational healthcare 11252:Differentiable computing 11171:Graphics processing unit 10597:Domain-specific language 10466:Computational complexity 10259:Recreational mathematics 9410:Arithmetic combinatorics 9170:Elementary Number Theory 9132:; Herbert S. Zuckerman; 8825:For other editions, see 8288:Long, Calvin T. (1972). 8102:; Wright, E.M. (2008) . 8007:"Analytic number theory" 7743:Colebrooke, Henry Thomas 7695:A History of Mathematics 6616:Tannery & Henry 1891 6592:Tannery & Henry 1891 6580:Tannery & Henry 1891 6548:Tannery & Henry 1891 6532:Tannery & Henry 1891 6497:Numbers and Measurements 6478:Tannery & Henry 1891 6378:Sachau & Bīrūni 1888 6372:, p. lxv, cited in 6308:, p. 325, cited in 5988:, p. 52) cited in: 5951:, p. 108. See also 5463:, p. xv); cited in 5439:, 1662) and France (the 5306:, or, what is the same, 5180:, Ch. 3, Problem 36, in 5146:, Ch. 3, Problem 26, in 5033: 5013:Algebraic function field 4596:arithmetic combinatorics 4443:Arithmetic combinatorics 4437:Arithmetic combinatorics 4067:{\displaystyle f(x,y)=0} 4021:{\displaystyle f(x,y)=0} 3980:{\displaystyle f(x,y)=0} 3911:{\displaystyle f(x,y)=0} 3080:in quadratic fields. (A 2660:Elementary number theory 2007: 1160:Diophantus of Alexandria 1141:Diophantus of Alexandria 39:Not to be confused with 18:Elementary number theory 11232:Computational chemistry 11166:Photograph manipulation 11057:Artificial intelligence 10873:Decision support system 10144:Mathematical statistics 10134:Mathematical psychology 10104:Engineering mathematics 10038:Algebraic number theory 9381:Geometric number theory 9337:Algebraic number theory 9085:Oxford University Press 9077:; E.M. Wright (2008) . 9003:Oxford University Press 8801:Taylor, Thomas (trans.) 8261:; Ang, Tian Se (2004). 8110:Oxford University Press 8078:Guthrie, Kenneth Sylvan 7856:Varia Opera Mathematica 6564:Varia Opera Mathematica 6396:, pp. 97–125, and 5943:,(trans., for example, 5766:, p. 40. The term 5648:to be complex numbers: 5242:and a non-zero integer 5090:Hardy & Wright 2008 4868:Hilbert's tenth problem 4825:{\displaystyle ax+by=c} 4788:greatest common divisor 3568:is to determine when a 3564:The central problem of 3056:algebraic number fields 2901:Algebraic number theory 2895:Algebraic number theory 2881:Dedekind zeta functions 2751:. The color of a point 2542:algebraic number theory 2037:Fermat's little theorem 1996:{\displaystyle n\geq 3} 1591:Fermat's little theorem 995:—that it is known that 470:{\displaystyle (a,b,c)} 423:The Plimpton 322 tablet 385:.) The use of the term 363:elementary calculations 297:. German mathematician 11297:Educational technology 11128:Reinforcement learning 10878:Process control system 10776:Computational geometry 10766:Algorithmic efficiency 10761:Analysis of algorithms 10416:Systems on Chip (SoCs) 10290:Mathematics portal 10139:Mathematical sociology 10119:Mathematical economics 10114:Mathematical chemistry 10043:Analytic number theory 9924:Differential equations 9500:Transcendental numbers 9414:additive number theory 9363:Analytic number theory 9249:A course in arithmetic 9019:. Boston: Birkhäuser. 8731:A Course in Arithmetic 8682:10.1006/hmat.2001.2317 8309:Mahoney, M.S. (1994). 8294:D.C. Heath and Company 8212:Analytic Number Theory 8190:(Fall 2011 ed.). 8053:Guthrie, K.S. (trans.) 8013:; Barrow-Green, June; 7305:Krishna, Hari (2017). 7174:. Vol. XXVIII.2. 6618:, Vol. I, pp. 340–341. 6215:Tusc. Disput. 1.17.39. 5998:Catalogue of Geometers 5864: 5718: 5634: 5388: 5351:, there is an integer 5276: 5200: 5167: 5126: 4942:Error correction codes 4926:fast Fourier transform 4826: 4772: 4732: 4712: 4692: 4658: 4642:additive combinatorics 4618: 4601:additive number theory 4585: 4565: 4488: 4468: 4451:Does a fairly "thick" 4447:Additive number theory 4415: 4376:emerged in the 1970s. 4308: 4288: 4268: 4248: 4184: 4164: 4123: 4095: 4068: 4022: 3981: 3932: 3912: 3873: 3851: 3805: 3771: 3737: 3676: 3644: 3442: 3409: 3376: 3356: 3336: 3269: 3237: 3217: 3171: 3151: 3131: 3111: 3047: 2962: 2942: 2941:{\displaystyle f(x)=0} 2856:(and, more generally, 2788:Analytic number theory 2784: 2764: 2732:Analytic number theory 2726:Analytic number theory 2718: 2580:) as a field of study. 2569: 2559: 2534: 2505: 2483: 2457: 2375: 2321: 2308: 2267: 2172: 2115: 2079: 2017: 1997: 1971: 1911: 1858: 1797: 1741: 1701: 1647: 1555: 1476: 1393: 1343: 1268: 1214: 1155: 1043: 968: 810: 777: 711: 676: 524: 471: 424: 336:). One may also study 334:analytic number theory 63: 11267:Electronic publishing 11237:Computational biology 11227:Computational physics 11123:Unsupervised learning 11037:Distributed computing 10913:Information retrieval 10820:Mathematical analysis 10810:Mathematical software 10700:Theory of computation 10665:Software construction 10655:Requirements analysis 10533:Software organization 10461:Computer architecture 10431:Hardware acceleration 10396:Printed circuit board 10269:Mathematics education 10199:Theory of computation 9919:Hypercomplex analysis 9571:Irrationality measure 9561:Diophantine equations 9404:Hodge–Arakelov theory 9144:John Wiley & Sons 8504:Sachs, Abraham Joseph 8100:Hardy, Godfrey Harold 7918:Gauss, Carl Friedrich 7665:(Subscription needed) 7430:10.1201/9780203756676 7418:Baylis, John (2018). 7271:10.1201/9781351664110 6302:Brāhmasphuṭasiddhānta 5865: 5719: 5635: 5441:Académie des sciences 5389: 5277: 5185: 5151: 5117: 5072:The Higher Arithmetic 4876:computably enumerable 4827: 4770:Diophantine equations 4755: 4733: 4713: 4693: 4659: 4619: 4586: 4566: 4489: 4469: 4416: 4324:transcendental number 4309: 4289: 4269: 4249: 4185: 4165: 4124: 4096: 4069: 4023: 3982: 3933: 3913: 3863: 3852: 3806: 3804:{\displaystyle y=b/c} 3772: 3770:{\displaystyle x=a/c} 3738: 3677: 3675:{\displaystyle (x,y)} 3645: 3454:quadratic reciprocity 3443: 3410: 3377: 3357: 3337: 3270: 3238: 3218: 3172: 3152: 3132: 3112: 3048: 2963: 2943: 2885:Riemann zeta function 2826:twin prime conjecture 2770: 2755:gives the value of ζ( 2741:Riemann zeta function 2739: 2708: 2565: 2555: 2530: 2514:quadratic reciprocity 2503: 2484: 2458: 2385:Adrien-Marie Legendre 2376: 2325:Joseph-Louis Lagrange 2319: 2305:Fermat's Last Theorem 2294: 2281:Diophantine equations 2275:quadratic reciprocity 2268: 2222:Riemann zeta function 2173: 2123:Joseph-Louis Lagrange 2116: 2080: 2015: 1998: 1972: 1922:Fermat's Last Theorem 1912: 1859: 1798: 1742: 1702: 1648: 1551: 1489:Brāhmasphuṭasiddhānta 1459: 1446:Further information: 1394: 1344: 1277:Diophantine equations 1269: 1215: 1138: 1044: 964: 929:Further information: 811: 778: 712: 677: 525: 472: 439:", that is, integers 422: 330:Riemann zeta function 49: 11027:Concurrent computing 10999:Ubiquitous computing 10971:Application security 10966:Information security 10795:Discrete mathematics 10771:Randomized algorithm 10723:Computability theory 10708:Model of computation 10680:Software maintenance 10675:Software engineering 10637:Software development 10587:Programming language 10582:Programming paradigm 10499:Network architecture 10249:Informal mathematics 10129:Mathematical physics 10124:Mathematical finance 10109:Mathematical biology 10048:Diophantine geometry 9530:Arithmetic functions 9396:Diophantine geometry 9208:Shafarevich, Igor R. 8753:Smith, D.E. (1958). 8670:Historia Mathematica 8589:Mathematics in India 7971:Goldstein, Catherine 7877:Historia Mathematica 7798:Mathematics Magazine 7686:Boyer, Carl Benjamin 7178:. pp. 323–378. 6924:From the preface of 6042:van der Waerden 1961 6018:van der Waerden 1961 5965:van der Waerden 1961 5949:van der Waerden 1961 5925:van der Waerden 1961 5884:van der Waerden 1961 5872:van der Waerden 1961 5790: 5652: 5598: 5359: 5250: 5223:van der Waerden 1961 5133:, pp. 199–200) 4898:discrete mathematics 4887:The number-theorist 4798: 4722: 4702: 4676: 4648: 4608: 4575: 4498: 4478: 4458: 4405: 4349:Diophantine geometry 4298: 4278: 4258: 4194: 4174: 4133: 4105: 4085: 4037: 3991: 3950: 3922: 3881: 3815: 3781: 3747: 3694: 3654: 3605: 3600:Pythagorean equation 3570:Diophantine equation 3566:Diophantine geometry 3560:Diophantine geometry 3554:Diophantine geometry 3419: 3386: 3366: 3346: 3279: 3247: 3227: 3200: 3161: 3141: 3121: 3088: 2972: 2952: 2917: 2871:(generalizations of 2818:prime number theorem 2676:prime number theorem 2504:Carl Friedrich Gauss 2489:(completing work by 2467: 2399: 2389:prime number theorem 2339: 2234: 2129: 2089: 2043: 1981: 1928: 1868: 1815: 1758: 1711: 1671: 1605: 1353: 1303: 1282:polynomial equations 1274:. Thus, nowadays, a 1224: 1176: 1090:infinitude of primes 1003: 796: 763: 722:Babylonian astronomy 693: 551: 481: 443: 322:Diophantine geometry 307:mathematical objects 299:Carl Friedrich Gauss 295:arithmetic functions 135:Discrete mathematics 50:The distribution of 11302:Document management 11292:Operations research 11217:Enterprise software 11133:Multi-task learning 11118:Supervised learning 10840:Information systems 10670:Software deployment 10627:Software repository 10481:Real-time computing 10264:Mathematics and art 10174:Operations research 9929:Functional analysis 9576:Continued fractions 9439:Arithmetic dynamics 9434:Arithmetic topology 9428:P-adic Hodge theory 9420:Arithmetic geometry 9353:Iwasawa–Tate theory 8861:Elements of algebra 8524:"Thales of Miletus" 8500:Neugebauer, Otto E. 8478:Neugebauer, Otto E. 8360:Montgomery, Hugh L. 7942:Goldfeld, Dorian M. 7768:Montgomery, Hugh L. 7751:. London: J. Murray 7389:2011RvMP...83..307S 7077:good approximations 7025:, pp. 322–348. 6903:, pp. 337–338. 6891:, pp. 332–334. 6879:, pp. 327–328. 6822:, pp. 179–181. 6798:, pp. 285–291. 6786:, pp. 177–179. 6722:, pp. 178–179. 6642:, pp. 115–116. 6416:, pp. 305–321. 6227:, pp. 305–319. 6124:, pp. 138–140. 6070:Spiral of Theodorus 5506:functional analysis 5184:, pp. 223–224: 5150:, pp. 219–220: 5062:Take, for example, 4902:computer scientists 4870:, that there is no 4857:encrypting messages 4691:{\displaystyle A+A} 4634:finite group theory 4626:geometry of numbers 4431:Cramér's conjecture 4353:arithmetic geometry 4345:Arithmetic geometry 4341:geometry of numbers 4122:{\displaystyle a/q} 3864:Two examples of an 2822:Goldbach conjecture 2674:. For example, the 2528:and number theory: 2482:{\displaystyle n=5} 2329:four-square theorem 2178:(implying the case 1082:Euclidean algorithm 867:early modern period 790:Pythagorean mystics 710:{\displaystyle c/a} 437:Pythagorean triples 69:Part of a series on 11085:Search methodology 11032:Parallel computing 10989:Interaction design 10898:Computing platform 10825:Numerical analysis 10815:Information theory 10607:Software framework 10570:Software notations 10509:Network components 10406:Integrated circuit 10209:Numerical analysis 9818:Mathematical logic 9813:Information theory 9520:Modular arithmetic 9490:Irrational numbers 9424:anabelian geometry 9341:class field theory 9244:Serre, Jean-Pierre 9134:Hugh L. Montgomery 9055:but not under the 8726:Serre, Jean-Pierre 8643:10.1007/BF00717654 8522:(September 2004). 8364:Vaughan, Robert C. 8058:Life of Pythagoras 7794:Edwards, Harold M. 7156:Matiyasevich, Yuri 7113:See, for example, 6710:, pp. 176–189 6654:, pp. 2, 172. 6594:, Vol. II, p. 423. 6582:, Vol. II, p. 213. 6135:Lam & Ang 2004 6122:Lam & Ang 2004 6110:Lam & Ang 2004 5959:, paragraph 6, in 5957:Life of Pythagoras 5941:Life of Pythagoras 5860: 5714: 5630: 5496:) rather than for 5451:, p. 163 and 5384: 5272: 5182:Lam & Ang 2004 5176:See, for example, 5148:Lam & Ang 2004 5005:Mathematics portal 4936:Riemann hypothesis 4906:numerical analysis 4822: 4773: 4728: 4708: 4688: 4654: 4614: 4581: 4561: 4484: 4464: 4411: 4357:Faltings's theorem 4304: 4284: 4264: 4244: 4180: 4160: 4119: 4091: 4064: 4018: 3977: 3942:of the curve. The 3928: 3908: 3874: 3847: 3801: 3767: 3733: 3672: 3640: 3529:class field theory 3438: 3405: 3372: 3352: 3332: 3265: 3233: 3213: 3167: 3147: 3127: 3107: 3043: 2958: 2938: 2838:Riemann hypothesis 2785: 2781:fundamental domain 2771:The action of the 2765: 2719: 2688:Tauberian theorems 2570: 2560: 2506: 2479: 2453: 2371: 2322: 2309: 2285:elliptic integrals 2263: 2206:pentagonal numbers 2168: 2111: 2075: 2018: 1993: 1967: 1907: 1854: 1793: 1737: 1697: 1643: 1556: 1477: 1452:Islamic Golden Age 1389: 1339: 1264: 1210: 1156: 1147:, translated into 1039: 937:Hellenistic period 863:pentagonal numbers 859:triangular numbers 806: 773: 733:elementary algebra 729:Babylonian algebra 707: 672: 520: 467: 433:Larsa, Mesopotamia 425: 415:Dawn of arithmetic 367:mathematical logic 315:algebraic integers 236:Mathematics Portal 64: 11348: 11347: 11277:Electronic voting 11207:Quantum Computing 11200:Applied computing 11186:Image compression 10956:Hardware security 10946:Security services 10903:Digital marketing 10690:Open-source model 10602:Modeling language 10514:Network scheduler 10335: 10334: 9934:Harmonic analysis 9721: 9720: 9642: 9641: 9539:Advanced concepts 9495:Algebraic numbers 9480:Composite numbers 9287:Number Theory Web 9263:978-0-387-90040-7 9227:978-0-12-117850-5 9184:978-0-321-71775-7 9175:Pearson Education 9153:978-81-265-1811-1 9094:978-0-19-921986-5 9026:978-0-8176-3141-3 8999:Science Awakening 8920:978-0-8218-3580-7 8889:978-0-88385-558-4 8854:978-0-387-96014-2 8776:Oeuvres de Fermat 8767:Fermat, Pierre de 8745:978-0-387-90040-7 8599:978-0-691-12067-6 8548:Ya'qub, ibn Tariq 8520:O'Grady, Patricia 8491:978-0-486-22332-2 8410:978-0-691-02090-7 8379:978-0-521-84903-6 8343:(18 March 2017). 8326:978-0-691-03666-3 8274:978-981-238-696-0 8221:978-0-8218-3633-0 8174:978-0-521-32763-3 8119:978-0-19-921986-5 8091:978-0-933999-51-0 8034:978-0-691-11880-2 8003:Granville, Andrew 7988:978-3-540-20441-1 7933:978-0-387-96254-2 7909:978-1-4020-0081-2 7851:Fermat, Pierre de 7842:978-0-387-95002-0 7785:978-0-387-95097-6 7764:Davenport, Harold 7709:978-0-471-54397-8 7640:978-0-486-44619-6 7615:978-0-387-90163-3 7584:978-0-691-11485-9 7487:978-1-4020-0005-8 7439:978-0-203-75667-6 7318:978-1-351-45497-1 7280:978-1-351-66411-0 7249:978-3-540-10382-0 7224:978-0-8218-5501-0 7185:978-0-8218-1428-4 6985:See the proof in 6854:. November 2000. 6810:, pp. 55–56. 6774:, pp. 44–47. 6706:, p. 39 and 6444:, pp. 45–46. 6304:, XVIII, 3–5 (in 6239:, pp. 17–24. 6137:, pp. 27–28. 5994:Eudemus of Rhodes 5963:Van der Waerden ( 5343:, that is, given 5045:Already in 1921, 4956:equal temperament 4850:testing primality 4768:and solve simple 4738:may be compared. 4731:{\displaystyle A} 4711:{\displaystyle A} 4657:{\displaystyle A} 4617:{\displaystyle A} 4584:{\displaystyle A} 4487:{\displaystyle a} 4467:{\displaystyle A} 4414:{\displaystyle 0} 4307:{\displaystyle x} 4287:{\displaystyle x} 4267:{\displaystyle c} 4242: 4183:{\displaystyle x} 4094:{\displaystyle x} 4081:: given a number 3931:{\displaystyle f} 3548:Langlands program 3463:is said to be an 3436: 3403: 3375:{\displaystyle 3} 3355:{\displaystyle 2} 3327: 3305: 3236:{\displaystyle 6} 3211: 3179:chakravala method 3170:{\displaystyle d} 3150:{\displaystyle b} 3130:{\displaystyle a} 3105: 2961:{\displaystyle x} 2865:algebraic numbers 2858:automorphic forms 2709:Number theorists 2655:Main subdivisions 2578:higher arithmetic 1469:Alhasen [ 1063:Euclid's Elements 1037: 1021: 1011: 804: 771: 752:Euclid's Elements 651: 630: 589: 568: 283:higher arithmetic 270: 269: 225: 224: 16:(Redirected from 11368: 11338: 11337: 11328: 11327: 11318: 11317: 11138:Cross-validation 11110:Machine learning 10994:Social computing 10961:Network security 10756:Algorithm design 10685:Programming team 10645:Control variable 10622:Software library 10560:Software quality 10555:Operating system 10504:Network protocol 10369:Computer science 10362: 10355: 10348: 10339: 10338: 10323: 10322: 10311: 10310: 10299: 10298: 10288: 10287: 10219:Computer algebra 10194:Computer science 9914:Complex analysis 9748: 9741: 9734: 9725: 9724: 9669: 9662: 9655: 9646: 9645: 9632: 9622: 9612: 9611: 9602: 9601: 9592: 9591: 9485:Rational numbers 9316: 9309: 9302: 9293: 9292: 9267: 9239: 9194: 9192: 9191: 9173:(6th ed.). 9163: 9161: 9160: 9116: 9109:Vinogradov, I.M. 9104: 9102: 9101: 9036: 9034: 9033: 9006: 8990: 8988: 8987: 8981: 8958: 8940: 8930: 8928: 8927: 8899: 8897: 8896: 8858: 8824: 8818: 8810: 8780: 8758: 8749: 8721: 8719: 8718: 8692: 8690: 8684:. Archived from 8667: 8654: 8625: 8623: 8622: 8603: 8579: 8559: 8539: 8537: 8535: 8515: 8495: 8473: 8471: 8470: 8464: 8441: 8428: 8422: 8414: 8389: 8387: 8386: 8355: 8353: 8351: 8336: 8334: 8333: 8305: 8284: 8282: 8281: 8254: 8252: 8251: 8225: 8203: 8201: 8199: 8178: 8155: 8153: 8152: 8136:Heath, Thomas L. 8131: 8108:(6th ed.). 8095: 8073: 8071: 8070: 8044: 8042: 8041: 7998: 7996: 7995: 7966: 7964: 7963: 7957: 7950: 7937: 7913: 7894: 7892: 7867: 7865: 7864: 7846: 7821: 7789: 7759: 7757: 7756: 7738: 7736: 7735: 7713: 7690:Merzbach, Uta C. 7681: 7664: 7643: 7625: 7623: 7622: 7587: 7559: 7558: 7540: 7508: 7502: 7501: 7500: 7499: 7461: 7455: 7454: 7452: 7451: 7415: 7409: 7408: 7382: 7362: 7356: 7355: 7349: 7341: 7339: 7338: 7302: 7296: 7295: 7293: 7292: 7257: 7251: 7241: 7235: 7232: 7226: 7213: 7207: 7204:Solomon Feferman 7197: 7164:Felix E. Browder 7148: 7142: 7136: 7130: 7124: 7118: 7111: 7105: 7099: 7093: 7086: 7080: 7070: 7064: 7057: 7051: 7045: 7039: 7032: 7026: 7020: 7014: 7011:Varadarajan 2006 7008: 7002: 6996: 6990: 6983: 6977: 6971: 6965: 6959: 6950: 6939: 6933: 6922: 6916: 6910: 6904: 6898: 6892: 6886: 6880: 6874: 6868: 6867: 6865: 6863: 6844: 6838: 6832: 6823: 6817: 6811: 6808:Varadarajan 2006 6805: 6799: 6793: 6787: 6781: 6775: 6772:Varadarajan 2006 6769: 6763: 6760:Varadarajan 2006 6757: 6751: 6745: 6739: 6736:Varadarajan 2006 6729: 6723: 6717: 6711: 6704:Varadarajan 2006 6701: 6695: 6692:Varadarajan 2006 6690:, p. 2 and 6685: 6679: 6673: 6667: 6664:Varadarajan 2006 6661: 6655: 6649: 6643: 6637: 6631: 6625: 6619: 6613: 6607: 6601: 6595: 6589: 6583: 6577: 6571: 6545: 6539: 6529: 6523: 6522: 6520: 6519: 6491: 6485: 6471: 6465: 6451: 6445: 6439: 6433: 6423: 6417: 6411: 6405: 6391: 6385: 6367: 6361: 6355: 6349: 6343: 6337: 6331: 6325: 6319: 6313: 6286: 6280: 6274: 6268: 6261: 6255: 6249: 6240: 6234: 6228: 6222: 6216: 6213: 6207: 6204: 6198: 6197: 6195: 6194: 6179: 6173: 6167: 6161: 6156: 6150: 6144: 6138: 6131: 6125: 6103: 6097: 6091: 6085: 6079: 6073: 6051: 6045: 6039: 6033: 6027: 6021: 6011: 6005: 5996:(now lost), the 5974: 5968: 5934: 5928: 5922: 5916: 5910: 5904: 5893: 5887: 5881: 5875: 5869: 5867: 5866: 5861: 5856: 5855: 5843: 5842: 5818: 5817: 5805: 5804: 5781: 5775: 5761: 5755: 5749: 5733: 5723: 5721: 5720: 5715: 5707: 5706: 5679: 5678: 5639: 5637: 5636: 5631: 5623: 5622: 5610: 5609: 5592: 5586: 5515: 5509: 5490: 5484: 5474: 5468: 5465:Varadarajan 2006 5453:Varadarajan 2006 5417: 5411: 5393: 5391: 5390: 5385: 5383: 5382: 5294:") to mean that 5286:is congruent to 5281: 5279: 5278: 5273: 5271: 5270: 5232: 5226: 5210: 5204: 5174: 5168: 5141: 5135: 5124:, p. 202). 5099: 5093: 5060: 5054: 5043: 5007: 5002: 5001: 4922:Computer science 4831: 4829: 4828: 4823: 4762:digital computer 4737: 4735: 4734: 4729: 4717: 4715: 4714: 4709: 4697: 4695: 4694: 4689: 4663: 4661: 4660: 4655: 4623: 4621: 4620: 4615: 4590: 4588: 4587: 4582: 4570: 4568: 4567: 4562: 4493: 4491: 4490: 4485: 4473: 4471: 4470: 4465: 4420: 4418: 4417: 4412: 4330: 4313: 4311: 4310: 4305: 4293: 4291: 4290: 4285: 4273: 4271: 4270: 4265: 4253: 4251: 4250: 4245: 4243: 4241: 4240: 4228: 4223: 4215: 4201: 4189: 4187: 4186: 4181: 4169: 4167: 4166: 4161: 4128: 4126: 4125: 4120: 4115: 4100: 4098: 4097: 4092: 4073: 4071: 4070: 4065: 4027: 4025: 4024: 4019: 3986: 3984: 3983: 3978: 3937: 3935: 3934: 3929: 3917: 3915: 3914: 3909: 3856: 3854: 3853: 3848: 3840: 3839: 3827: 3826: 3810: 3808: 3807: 3802: 3797: 3776: 3774: 3773: 3768: 3763: 3742: 3740: 3739: 3734: 3732: 3731: 3719: 3718: 3706: 3705: 3681: 3679: 3678: 3673: 3649: 3647: 3646: 3641: 3630: 3629: 3617: 3616: 3487:, and the reals 3447: 3445: 3444: 3439: 3437: 3429: 3414: 3412: 3411: 3406: 3404: 3396: 3381: 3379: 3378: 3373: 3361: 3359: 3358: 3353: 3341: 3339: 3338: 3333: 3328: 3320: 3306: 3298: 3274: 3272: 3271: 3266: 3242: 3240: 3239: 3234: 3222: 3220: 3219: 3214: 3212: 3204: 3194:valuation theory 3190:theory of ideals 3176: 3174: 3173: 3168: 3156: 3154: 3153: 3148: 3136: 3134: 3133: 3128: 3116: 3114: 3113: 3108: 3106: 3101: 3052: 3050: 3049: 3044: 3030: 3029: 3014: 3013: 2998: 2984: 2983: 2967: 2965: 2964: 2959: 2947: 2945: 2944: 2939: 2907:algebraic number 2777:upper half plane 2696:Fourier analysis 2672:complex analysis 2633:Bernhard Riemann 2612:abstract algebra 2585:complex analysis 2488: 2486: 2485: 2480: 2462: 2460: 2459: 2454: 2446: 2445: 2430: 2429: 2414: 2413: 2380: 2378: 2377: 2372: 2370: 2369: 2354: 2353: 2272: 2270: 2269: 2264: 2262: 2261: 2246: 2245: 2214:complex analysis 2177: 2175: 2174: 2169: 2167: 2166: 2154: 2153: 2141: 2140: 2120: 2118: 2117: 2112: 2110: 2109: 2084: 2082: 2081: 2076: 2074: 2073: 2061: 2060: 2020:The interest of 2002: 2000: 1999: 1994: 1976: 1974: 1973: 1968: 1966: 1965: 1953: 1952: 1940: 1939: 1920:Fermat claimed ( 1916: 1914: 1913: 1908: 1906: 1905: 1893: 1892: 1880: 1879: 1863: 1861: 1860: 1855: 1853: 1852: 1840: 1839: 1827: 1826: 1811:(Obs. XLV) that 1802: 1800: 1799: 1794: 1786: 1785: 1770: 1769: 1746: 1744: 1743: 1738: 1736: 1735: 1723: 1722: 1706: 1704: 1703: 1698: 1696: 1695: 1683: 1682: 1652: 1650: 1649: 1644: 1639: 1638: 1623: 1622: 1578:amicable numbers 1559:Pierre de Fermat 1553:Pierre de Fermat 1513:Wilson's theorem 1436:Henry Colebrooke 1398: 1396: 1395: 1390: 1388: 1387: 1382: 1381: 1371: 1370: 1348: 1346: 1345: 1340: 1338: 1337: 1332: 1331: 1321: 1320: 1273: 1271: 1270: 1265: 1263: 1262: 1219: 1217: 1216: 1211: 1209: 1208: 1096:, Prop. IX.20). 1065:is described by 1055:incommensurables 1049:are irrational. 1048: 1046: 1045: 1040: 1038: 1033: 1022: 1017: 1012: 1007: 999:had proven that 949:, respectively. 903: 902: 847:figurate numbers 815: 813: 812: 807: 805: 800: 782: 780: 779: 774: 772: 767: 750:In book nine of 716: 714: 713: 708: 703: 681: 679: 678: 673: 668: 667: 662: 658: 657: 653: 652: 644: 631: 623: 606: 605: 600: 596: 595: 591: 590: 582: 569: 561: 529: 527: 526: 521: 519: 518: 506: 505: 493: 492: 476: 474: 473: 468: 399:number-theoretic 377:computer science 372:Peano arithmetic 311:rational numbers 287:pure mathematics 262: 255: 248: 234: 98: 97: 66: 65: 21: 11376: 11375: 11371: 11370: 11369: 11367: 11366: 11365: 11351: 11350: 11349: 11344: 11335: 11306: 11287:Word processing 11195: 11181:Virtual reality 11142: 11104: 11075:Computer vision 11051: 11047:Multiprocessing 11013: 10975: 10941:Security hacker 10917: 10893:Digital library 10834: 10785:Mathematics of 10780: 10742: 10718:Automata theory 10713:Formal language 10694: 10660:Software design 10631: 10564: 10550:Virtual machine 10528: 10524:Network service 10485: 10476:Embedded system 10449: 10382: 10371: 10366: 10336: 10331: 10282: 10273: 10223: 10180: 10159:Systems science 10090: 10086:Homotopy theory 10052: 10019: 9971: 9943: 9890: 9837: 9808:Category theory 9794: 9759: 9752: 9722: 9717: 9679: 9673: 9643: 9638: 9580: 9546:Quadratic forms 9534: 9509:P-adic analysis 9465:Natural numbers 9443: 9400:Arakelov theory 9325: 9320: 9274: 9264: 9242: 9228: 9204:Borevich, A. I. 9189: 9187: 9185: 9158: 9156: 9154: 9099: 9097: 9095: 9067: 9065:Further reading 9039: 9031: 9029: 9027: 8985: 8983: 8979: 8965:10.2307/2589706 8938: 8925: 8923: 8921: 8894: 8892: 8890: 8868:Truesdell, C.A. 8855: 8845:Springer-Verlag 8833:Truesdell, C.A. 8812: 8811: 8791:Volume 4 (1912) 8746: 8716: 8714: 8688: 8665: 8659:Robson, Eleanor 8620: 8618: 8600: 8533: 8531: 8492: 8468: 8466: 8462: 8439: 8416: 8415: 8411: 8384: 8382: 8380: 8349: 8347: 8331: 8329: 8327: 8279: 8277: 8275: 8249: 8247: 8222: 8208:Iwaniec, Henryk 8197: 8195: 8175: 8150: 8148: 8146:Clarendon Press 8120: 8092: 8068: 8066: 8039: 8037: 8035: 8011:Gowers, Timothy 7993: 7991: 7989: 7961: 7959: 7955: 7948: 7934: 7910: 7862: 7860: 7843: 7833:Springer Verlag 7810:10.2307/2690368 7786: 7754: 7752: 7733: 7731: 7710: 7641: 7620: 7618: 7616: 7592:Apostol, Tom M. 7585: 7568: 7563: 7562: 7509: 7505: 7497: 7495: 7488: 7462: 7458: 7449: 7447: 7440: 7416: 7412: 7363: 7359: 7343: 7342: 7336: 7334: 7319: 7303: 7299: 7290: 7288: 7281: 7259: 7258: 7254: 7242: 7238: 7233: 7229: 7214: 7210: 7186: 7160:Robinson, Julia 7149: 7145: 7137: 7133: 7125: 7121: 7112: 7108: 7100: 7096: 7087: 7083: 7071: 7067: 7058: 7054: 7046: 7042: 7033: 7029: 7021: 7017: 7009: 7005: 6997: 6993: 6984: 6980: 6972: 6968: 6960: 6953: 6940: 6936: 6923: 6919: 6911: 6907: 6899: 6895: 6887: 6883: 6875: 6871: 6861: 6859: 6846: 6845: 6841: 6833: 6826: 6818: 6814: 6806: 6802: 6794: 6790: 6782: 6778: 6770: 6766: 6758: 6754: 6746: 6742: 6730: 6726: 6718: 6714: 6702: 6698: 6686: 6682: 6678:, pp. 1–2. 6674: 6670: 6662: 6658: 6650: 6646: 6638: 6634: 6626: 6622: 6614: 6610: 6602: 6598: 6590: 6586: 6578: 6574: 6558:are taken from 6546: 6542: 6530: 6526: 6517: 6515: 6508: 6492: 6488: 6472: 6468: 6452: 6448: 6440: 6436: 6424: 6420: 6412: 6408: 6392: 6388: 6370:Colebrooke 1817 6368: 6364: 6358:Colebrooke 1817 6356: 6352: 6344: 6340: 6332: 6328: 6320: 6316: 6306:Colebrooke 1817 6287: 6283: 6275: 6271: 6262: 6258: 6250: 6243: 6235: 6231: 6223: 6219: 6214: 6210: 6205: 6201: 6192: 6190: 6181: 6180: 6176: 6168: 6164: 6157: 6153: 6145: 6141: 6132: 6128: 6104: 6100: 6092: 6088: 6080: 6076: 6052: 6048: 6040: 6036: 6028: 6024: 6012: 6008: 5980:28), cited in: 5975: 5971: 5935: 5931: 5923: 5919: 5911: 5907: 5901:Neugebauer 1969 5897:Neugebauer 1969 5894: 5890: 5882: 5878: 5851: 5847: 5838: 5834: 5813: 5809: 5800: 5796: 5791: 5788: 5787: 5782: 5778: 5762: 5758: 5750: 5746: 5741: 5736: 5702: 5698: 5674: 5670: 5653: 5650: 5649: 5618: 5614: 5605: 5601: 5599: 5596: 5595: 5593: 5589: 5559:Évariste Galois 5516: 5512: 5491: 5487: 5475: 5471: 5418: 5414: 5378: 5374: 5360: 5357: 5356: 5266: 5262: 5251: 5248: 5247: 5233: 5229: 5211: 5207: 5192: 5175: 5171: 5159: 5142: 5138: 5100: 5096: 5061: 5057: 5044: 5040: 5036: 5003: 4996: 4993: 4969: 4889:Leonard Dickson 4885: 4799: 4796: 4795: 4775:While the word 4750: 4744: 4723: 4720: 4719: 4703: 4700: 4699: 4677: 4674: 4673: 4649: 4646: 4645: 4609: 4606: 4605: 4576: 4573: 4572: 4499: 4496: 4495: 4479: 4476: 4475: 4459: 4456: 4455: 4449: 4441:Main articles: 4439: 4406: 4403: 4402: 4388: 4382: 4365: 4363:Other subfields 4328: 4299: 4296: 4295: 4279: 4276: 4275: 4259: 4256: 4255: 4236: 4232: 4227: 4219: 4211: 4197: 4195: 4192: 4191: 4175: 4172: 4171: 4134: 4131: 4130: 4111: 4106: 4103: 4102: 4086: 4083: 4082: 4038: 4035: 4034: 3992: 3989: 3988: 3951: 3948: 3947: 3923: 3920: 3919: 3882: 3879: 3878: 3835: 3831: 3822: 3818: 3816: 3813: 3812: 3793: 3782: 3779: 3778: 3759: 3748: 3745: 3744: 3727: 3723: 3714: 3710: 3701: 3697: 3695: 3692: 3691: 3655: 3652: 3651: 3625: 3621: 3612: 3608: 3606: 3603: 3602: 3593:integral points 3589:rational points 3562: 3556: 3428: 3420: 3417: 3416: 3395: 3387: 3384: 3383: 3367: 3364: 3363: 3347: 3344: 3343: 3319: 3297: 3280: 3277: 3276: 3248: 3245: 3244: 3228: 3225: 3224: 3203: 3201: 3198: 3197: 3162: 3159: 3158: 3142: 3139: 3138: 3122: 3119: 3118: 3100: 3089: 3086: 3085: 3082:quadratic field 3025: 3021: 3009: 3005: 2994: 2979: 2975: 2973: 2970: 2969: 2953: 2950: 2949: 2918: 2915: 2914: 2903: 2897: 2877:can be answered 2790:may be defined 2734: 2728: 2662: 2657: 2550: 2538:Évariste Galois 2468: 2465: 2464: 2441: 2437: 2425: 2421: 2409: 2405: 2400: 2397: 2396: 2365: 2361: 2349: 2345: 2340: 2337: 2336: 2333:quadratic forms 2314: 2257: 2253: 2241: 2237: 2235: 2232: 2231: 2228:Quadratic forms 2191:Pell's equation 2162: 2158: 2149: 2145: 2136: 2132: 2130: 2127: 2126: 2105: 2101: 2090: 2087: 2086: 2085:if and only if 2069: 2065: 2056: 2052: 2044: 2041: 2040: 2010: 1982: 1979: 1978: 1961: 1957: 1948: 1944: 1935: 1931: 1929: 1926: 1925: 1901: 1897: 1888: 1884: 1875: 1871: 1869: 1866: 1865: 1848: 1844: 1835: 1831: 1822: 1818: 1816: 1813: 1812: 1781: 1777: 1765: 1761: 1759: 1756: 1755: 1731: 1727: 1718: 1714: 1712: 1709: 1708: 1691: 1687: 1678: 1674: 1672: 1669: 1668: 1634: 1630: 1612: 1608: 1606: 1603: 1602: 1570:perfect numbers 1546: 1541: 1521: 1454: 1444: 1383: 1377: 1373: 1372: 1366: 1362: 1354: 1351: 1350: 1333: 1327: 1323: 1322: 1316: 1312: 1304: 1301: 1300: 1290: 1258: 1254: 1225: 1222: 1221: 1204: 1200: 1177: 1174: 1173: 1133: 1121:Pell's equation 1032: 1016: 1006: 1004: 1001: 1000: 933: 927: 916:Alexander Wylie 799: 797: 794: 793: 766: 764: 761: 760: 699: 694: 691: 690: 663: 643: 636: 632: 622: 621: 617: 616: 601: 581: 574: 570: 560: 559: 555: 554: 552: 549: 548: 514: 510: 501: 497: 488: 484: 482: 479: 478: 444: 441: 440: 417: 412: 407: 266: 221: 220: 171: 163: 162: 158:Decision theory 106: 44: 37: 28: 23: 22: 15: 12: 11: 5: 11374: 11364: 11363: 11346: 11345: 11343: 11342: 11332: 11322: 11311: 11308: 11307: 11305: 11304: 11299: 11294: 11289: 11284: 11279: 11274: 11269: 11264: 11259: 11254: 11249: 11244: 11239: 11234: 11229: 11224: 11219: 11214: 11209: 11203: 11201: 11197: 11196: 11194: 11193: 11191:Solid modeling 11188: 11183: 11178: 11173: 11168: 11163: 11158: 11152: 11150: 11144: 11143: 11141: 11140: 11135: 11130: 11125: 11120: 11114: 11112: 11106: 11105: 11103: 11102: 11097: 11092: 11090:Control method 11087: 11082: 11077: 11072: 11067: 11061: 11059: 11053: 11052: 11050: 11049: 11044: 11042:Multithreading 11039: 11034: 11029: 11023: 11021: 11015: 11014: 11012: 11011: 11006: 11001: 10996: 10991: 10985: 10983: 10977: 10976: 10974: 10973: 10968: 10963: 10958: 10953: 10948: 10943: 10938: 10936:Formal methods 10933: 10927: 10925: 10919: 10918: 10916: 10915: 10910: 10908:World Wide Web 10905: 10900: 10895: 10890: 10885: 10880: 10875: 10870: 10865: 10860: 10855: 10850: 10844: 10842: 10836: 10835: 10833: 10832: 10827: 10822: 10817: 10812: 10807: 10802: 10797: 10791: 10789: 10782: 10781: 10779: 10778: 10773: 10768: 10763: 10758: 10752: 10750: 10744: 10743: 10741: 10740: 10735: 10730: 10725: 10720: 10715: 10710: 10704: 10702: 10696: 10695: 10693: 10692: 10687: 10682: 10677: 10672: 10667: 10662: 10657: 10652: 10647: 10641: 10639: 10633: 10632: 10630: 10629: 10624: 10619: 10614: 10609: 10604: 10599: 10594: 10589: 10584: 10578: 10576: 10566: 10565: 10563: 10562: 10557: 10552: 10547: 10542: 10536: 10534: 10530: 10529: 10527: 10526: 10521: 10516: 10511: 10506: 10501: 10495: 10493: 10487: 10486: 10484: 10483: 10478: 10473: 10468: 10463: 10457: 10455: 10451: 10450: 10448: 10447: 10438: 10433: 10428: 10423: 10418: 10413: 10408: 10403: 10398: 10392: 10390: 10384: 10383: 10376: 10373: 10372: 10365: 10364: 10357: 10350: 10342: 10333: 10332: 10330: 10329: 10317: 10305: 10293: 10278: 10275: 10274: 10272: 10271: 10266: 10261: 10256: 10251: 10246: 10245: 10244: 10237:Mathematicians 10233: 10231: 10229:Related topics 10225: 10224: 10222: 10221: 10216: 10211: 10206: 10201: 10196: 10190: 10188: 10182: 10181: 10179: 10178: 10177: 10176: 10171: 10166: 10164:Control theory 10156: 10151: 10146: 10141: 10136: 10131: 10126: 10121: 10116: 10111: 10106: 10100: 10098: 10092: 10091: 10089: 10088: 10083: 10078: 10073: 10068: 10062: 10060: 10054: 10053: 10051: 10050: 10045: 10040: 10035: 10029: 10027: 10021: 10020: 10018: 10017: 10012: 10007: 10002: 9997: 9992: 9987: 9981: 9979: 9973: 9972: 9970: 9969: 9964: 9959: 9953: 9951: 9945: 9944: 9942: 9941: 9939:Measure theory 9936: 9931: 9926: 9921: 9916: 9911: 9906: 9900: 9898: 9892: 9891: 9889: 9888: 9883: 9878: 9873: 9868: 9863: 9858: 9853: 9847: 9845: 9839: 9838: 9836: 9835: 9830: 9825: 9820: 9815: 9810: 9804: 9802: 9796: 9795: 9793: 9792: 9787: 9782: 9781: 9780: 9775: 9764: 9761: 9760: 9751: 9750: 9743: 9736: 9728: 9719: 9718: 9716: 9715: 9710: 9705: 9700: 9695: 9690: 9684: 9681: 9680: 9672: 9671: 9664: 9657: 9649: 9640: 9639: 9637: 9636: 9626: 9616: 9606: 9604:List of topics 9596: 9585: 9582: 9581: 9579: 9578: 9573: 9568: 9563: 9558: 9553: 9548: 9542: 9540: 9536: 9535: 9533: 9532: 9527: 9522: 9517: 9512: 9505:P-adic numbers 9502: 9497: 9492: 9487: 9482: 9477: 9472: 9467: 9462: 9457: 9451: 9449: 9445: 9444: 9442: 9441: 9436: 9431: 9417: 9407: 9393: 9388: 9383: 9378: 9360: 9349:Iwasawa theory 9333: 9331: 9327: 9326: 9319: 9318: 9311: 9304: 9296: 9290: 9289: 9284: 9273: 9272:External links 9270: 9269: 9268: 9262: 9240: 9226: 9218:Academic Press 9196: 9195: 9183: 9164: 9152: 9118: 9117: 9105: 9093: 9066: 9063: 9062: 9061: 9038: 9037: 9025: 9007: 8991: 8956:10.1.1.383.545 8949:(4): 305–319. 8931: 8919: 8900: 8888: 8864: 8853: 8829: 8793: 8759: 8750: 8744: 8722: 8697:Sachau, Eduard 8693: 8691:on 2014-10-21. 8676:(3): 167–206. 8655: 8637:(4): 305–321. 8626: 8604: 8598: 8580: 8560: 8544:Pingree, David 8540: 8516: 8496: 8490: 8474: 8435:(March 2010). 8433:Mumford, David 8429: 8409: 8390: 8378: 8356: 8337: 8325: 8306: 8285: 8273: 8255: 8226: 8220: 8204: 8179: 8173: 8156: 8132: 8118: 8096: 8090: 8074: 8045: 8033: 7999: 7987: 7967: 7938: 7932: 7914: 7908: 7895: 7883:(3): 277–318. 7868: 7847: 7841: 7822: 7804:(5): 285–291. 7790: 7784: 7760: 7739: 7718: 7717:at archive.org 7708: 7682: 7666: 7644: 7639: 7626: 7614: 7588: 7583: 7569: 7567: 7564: 7561: 7560: 7503: 7486: 7456: 7438: 7410: 7373:(2): 307–330. 7357: 7317: 7297: 7279: 7252: 7236: 7227: 7208: 7184: 7143: 7131: 7119: 7106: 7102:Granville 2008 7094: 7081: 7073:Granville 2008 7065: 7052: 7040: 7027: 7023:Granville 2008 7015: 7003: 6991: 6978: 6966: 6951: 6949:, p. 25). 6934: 6917: 6905: 6893: 6881: 6869: 6839: 6837:, p. 181. 6824: 6812: 6800: 6788: 6776: 6764: 6752: 6750:, p. 183. 6740: 6724: 6712: 6696: 6680: 6668: 6656: 6644: 6632: 6630:, p. 115. 6620: 6608: 6596: 6584: 6572: 6540: 6524: 6507:978-1538300428 6506: 6486: 6466: 6464:, p. 33). 6446: 6434: 6418: 6406: 6404:, p. 256. 6386: 6384:, pp. 168 6362: 6350: 6348:, p. 283. 6338: 6336:, p. 194. 6326: 6324:, p. 388. 6314: 6312:, p. 42). 6281: 6279:, p. 387. 6269: 6267:, p. 42). 6256: 6254:, p. 119. 6241: 6229: 6217: 6208: 6199: 6174: 6162: 6159:Libbrecht 1973 6151: 6139: 6126: 6106:Sunzi Suanjing 6098: 6086: 6082:von Fritz 2004 6074: 6063:von Fritz 2004 6046: 6044:, p. 109. 6034: 6022: 6020:, p. 108. 6006: 5969: 5929: 5917: 5915:, p. 302. 5905: 5903:, p. 39). 5888: 5886:, p. 184. 5876: 5859: 5854: 5850: 5846: 5841: 5837: 5833: 5830: 5827: 5824: 5821: 5816: 5812: 5808: 5803: 5799: 5795: 5776: 5756: 5743: 5742: 5740: 5737: 5735: 5734: 5713: 5710: 5705: 5701: 5697: 5694: 5691: 5688: 5685: 5682: 5677: 5673: 5669: 5666: 5663: 5660: 5657: 5629: 5626: 5621: 5617: 5613: 5608: 5604: 5587: 5510: 5485: 5469: 5461:Truesdell 1984 5412: 5381: 5377: 5373: 5370: 5367: 5364: 5269: 5265: 5261: 5258: 5255: 5227: 5205: 5178:Sunzi Suanjing 5169: 5144:Sunzi Suanjing 5136: 5129:mathematics".( 5094: 5055: 5037: 5035: 5032: 5031: 5030: 5025: 5020: 5015: 5009: 5008: 4992: 4989: 4968: 4965: 4964: 4963: 4952: 4945: 4939: 4929: 4919: 4884: 4881: 4872:Turing machine 4859:(for example, 4821: 4818: 4815: 4812: 4809: 4806: 4803: 4760:, a primitive 4746:Main article: 4743: 4740: 4727: 4707: 4687: 4684: 4681: 4653: 4630:ergodic theory 4613: 4580: 4560: 4557: 4554: 4551: 4548: 4545: 4542: 4539: 4536: 4533: 4530: 4527: 4524: 4521: 4518: 4515: 4512: 4509: 4506: 4503: 4483: 4463: 4438: 4435: 4410: 4384:Main article: 4381: 4378: 4364: 4361: 4303: 4283: 4263: 4239: 4235: 4231: 4226: 4222: 4218: 4214: 4210: 4207: 4204: 4200: 4179: 4159: 4156: 4153: 4150: 4147: 4144: 4141: 4138: 4118: 4114: 4110: 4090: 4063: 4060: 4057: 4054: 4051: 4048: 4045: 4042: 4017: 4014: 4011: 4008: 4005: 4002: 3999: 3996: 3976: 3973: 3970: 3967: 3964: 3961: 3958: 3955: 3927: 3907: 3904: 3901: 3898: 3895: 3892: 3889: 3886: 3866:elliptic curve 3846: 3843: 3838: 3834: 3830: 3825: 3821: 3800: 3796: 3792: 3789: 3786: 3766: 3762: 3758: 3755: 3752: 3730: 3726: 3722: 3717: 3713: 3709: 3704: 3700: 3671: 3668: 3665: 3662: 3659: 3639: 3636: 3633: 3628: 3624: 3620: 3615: 3611: 3558:Main article: 3555: 3552: 3544:Iwasawa theory 3503:such that the 3435: 3432: 3427: 3424: 3402: 3399: 3394: 3391: 3371: 3351: 3331: 3326: 3323: 3318: 3315: 3312: 3309: 3304: 3301: 3296: 3293: 3290: 3287: 3284: 3264: 3261: 3258: 3255: 3252: 3232: 3210: 3207: 3166: 3146: 3126: 3104: 3099: 3096: 3093: 3042: 3039: 3036: 3033: 3028: 3024: 3020: 3017: 3012: 3008: 3004: 3001: 2997: 2993: 2990: 2987: 2982: 2978: 2957: 2937: 2934: 2931: 2928: 2925: 2922: 2911:complex number 2899:Main article: 2896: 2893: 2834:Waring problem 2807: 2806: 2803: 2730:Main article: 2727: 2724: 2692:Wiener–Ikehara 2690:(for example, 2661: 2658: 2656: 2653: 2635:(1859) on the 2604: 2603: 2596: 2581: 2549: 2546: 2526:roots of unity 2522:Disquisitiones 2495:Sophie Germain 2478: 2475: 2472: 2452: 2449: 2444: 2440: 2436: 2433: 2428: 2424: 2420: 2417: 2412: 2408: 2404: 2368: 2364: 2360: 2357: 2352: 2348: 2344: 2313: 2310: 2289: 2288: 2278: 2260: 2256: 2252: 2249: 2244: 2240: 2225: 2195: 2187: 2165: 2161: 2157: 2152: 2148: 2144: 2139: 2135: 2108: 2104: 2100: 2097: 2094: 2072: 2068: 2064: 2059: 2055: 2051: 2048: 2035:This includes 2022:Leonhard Euler 2016:Leonhard Euler 2009: 2006: 2005: 2004: 1992: 1989: 1986: 1964: 1960: 1956: 1951: 1947: 1943: 1938: 1934: 1918: 1904: 1900: 1896: 1891: 1887: 1883: 1878: 1874: 1851: 1847: 1843: 1838: 1834: 1830: 1825: 1821: 1805: 1792: 1789: 1784: 1780: 1776: 1773: 1768: 1764: 1752: 1734: 1730: 1726: 1721: 1717: 1694: 1690: 1686: 1681: 1677: 1653: 1642: 1637: 1633: 1629: 1626: 1621: 1618: 1615: 1611: 1588: 1585: 1545: 1542: 1540: 1537: 1520: 1517: 1509:Ibn al-Haytham 1497:Qusta ibn Luqa 1443: 1440: 1386: 1380: 1376: 1369: 1365: 1361: 1358: 1336: 1330: 1326: 1319: 1315: 1311: 1308: 1289: 1286: 1261: 1257: 1253: 1250: 1247: 1244: 1241: 1238: 1235: 1232: 1229: 1207: 1203: 1199: 1196: 1193: 1190: 1187: 1184: 1181: 1132: 1129: 1059:number systems 1036: 1031: 1028: 1025: 1020: 1015: 1010: 926: 923: 879:Sunzi Suanjing 851:square numbers 803: 770: 706: 702: 698: 687:Old Babylonian 683: 682: 671: 666: 661: 656: 650: 647: 642: 639: 635: 629: 626: 620: 615: 612: 609: 604: 599: 594: 588: 585: 580: 577: 573: 567: 564: 558: 517: 513: 509: 504: 500: 496: 491: 487: 466: 463: 460: 457: 454: 451: 448: 416: 413: 411: 408: 406: 403: 268: 267: 265: 264: 257: 250: 242: 239: 238: 227: 226: 223: 222: 219: 218: 213: 208: 203: 198: 193: 188: 183: 178: 172: 169: 168: 165: 164: 161: 160: 151: 146: 137: 132: 123: 118: 113: 107: 102: 101: 94: 93: 92: 91: 86: 78: 77: 71: 70: 26: 9: 6: 4: 3: 2: 11373: 11362: 11361:Number theory 11359: 11358: 11356: 11341: 11333: 11331: 11323: 11321: 11313: 11312: 11309: 11303: 11300: 11298: 11295: 11293: 11290: 11288: 11285: 11283: 11280: 11278: 11275: 11273: 11270: 11268: 11265: 11263: 11260: 11258: 11255: 11253: 11250: 11248: 11245: 11243: 11240: 11238: 11235: 11233: 11230: 11228: 11225: 11223: 11220: 11218: 11215: 11213: 11210: 11208: 11205: 11204: 11202: 11198: 11192: 11189: 11187: 11184: 11182: 11179: 11177: 11176:Mixed reality 11174: 11172: 11169: 11167: 11164: 11162: 11159: 11157: 11154: 11153: 11151: 11149: 11145: 11139: 11136: 11134: 11131: 11129: 11126: 11124: 11121: 11119: 11116: 11115: 11113: 11111: 11107: 11101: 11098: 11096: 11093: 11091: 11088: 11086: 11083: 11081: 11078: 11076: 11073: 11071: 11068: 11066: 11063: 11062: 11060: 11058: 11054: 11048: 11045: 11043: 11040: 11038: 11035: 11033: 11030: 11028: 11025: 11024: 11022: 11020: 11016: 11010: 11009:Accessibility 11007: 11005: 11004:Visualization 11002: 11000: 10997: 10995: 10992: 10990: 10987: 10986: 10984: 10982: 10978: 10972: 10969: 10967: 10964: 10962: 10959: 10957: 10954: 10952: 10949: 10947: 10944: 10942: 10939: 10937: 10934: 10932: 10929: 10928: 10926: 10924: 10920: 10914: 10911: 10909: 10906: 10904: 10901: 10899: 10896: 10894: 10891: 10889: 10886: 10884: 10881: 10879: 10876: 10874: 10871: 10869: 10866: 10864: 10861: 10859: 10856: 10854: 10851: 10849: 10846: 10845: 10843: 10841: 10837: 10831: 10828: 10826: 10823: 10821: 10818: 10816: 10813: 10811: 10808: 10806: 10803: 10801: 10798: 10796: 10793: 10792: 10790: 10788: 10783: 10777: 10774: 10772: 10769: 10767: 10764: 10762: 10759: 10757: 10754: 10753: 10751: 10749: 10745: 10739: 10736: 10734: 10731: 10729: 10726: 10724: 10721: 10719: 10716: 10714: 10711: 10709: 10706: 10705: 10703: 10701: 10697: 10691: 10688: 10686: 10683: 10681: 10678: 10676: 10673: 10671: 10668: 10666: 10663: 10661: 10658: 10656: 10653: 10651: 10648: 10646: 10643: 10642: 10640: 10638: 10634: 10628: 10625: 10623: 10620: 10618: 10615: 10613: 10610: 10608: 10605: 10603: 10600: 10598: 10595: 10593: 10590: 10588: 10585: 10583: 10580: 10579: 10577: 10575: 10571: 10567: 10561: 10558: 10556: 10553: 10551: 10548: 10546: 10543: 10541: 10538: 10537: 10535: 10531: 10525: 10522: 10520: 10517: 10515: 10512: 10510: 10507: 10505: 10502: 10500: 10497: 10496: 10494: 10492: 10488: 10482: 10479: 10477: 10474: 10472: 10471:Dependability 10469: 10467: 10464: 10462: 10459: 10458: 10456: 10452: 10446: 10442: 10439: 10437: 10434: 10432: 10429: 10427: 10424: 10422: 10419: 10417: 10414: 10412: 10409: 10407: 10404: 10402: 10399: 10397: 10394: 10393: 10391: 10389: 10385: 10380: 10374: 10370: 10363: 10358: 10356: 10351: 10349: 10344: 10343: 10340: 10328: 10327: 10318: 10316: 10315: 10306: 10304: 10303: 10294: 10292: 10291: 10286: 10280: 10279: 10276: 10270: 10267: 10265: 10262: 10260: 10257: 10255: 10252: 10250: 10247: 10243: 10240: 10239: 10238: 10235: 10234: 10232: 10230: 10226: 10220: 10217: 10215: 10212: 10210: 10207: 10205: 10202: 10200: 10197: 10195: 10192: 10191: 10189: 10187: 10186:Computational 10183: 10175: 10172: 10170: 10167: 10165: 10162: 10161: 10160: 10157: 10155: 10152: 10150: 10147: 10145: 10142: 10140: 10137: 10135: 10132: 10130: 10127: 10125: 10122: 10120: 10117: 10115: 10112: 10110: 10107: 10105: 10102: 10101: 10099: 10097: 10093: 10087: 10084: 10082: 10079: 10077: 10074: 10072: 10069: 10067: 10064: 10063: 10061: 10059: 10055: 10049: 10046: 10044: 10041: 10039: 10036: 10034: 10031: 10030: 10028: 10026: 10025:Number theory 10022: 10016: 10013: 10011: 10008: 10006: 10003: 10001: 9998: 9996: 9993: 9991: 9988: 9986: 9983: 9982: 9980: 9978: 9974: 9968: 9965: 9963: 9960: 9958: 9957:Combinatorics 9955: 9954: 9952: 9950: 9946: 9940: 9937: 9935: 9932: 9930: 9927: 9925: 9922: 9920: 9917: 9915: 9912: 9910: 9909:Real analysis 9907: 9905: 9902: 9901: 9899: 9897: 9893: 9887: 9884: 9882: 9879: 9877: 9874: 9872: 9869: 9867: 9864: 9862: 9859: 9857: 9854: 9852: 9849: 9848: 9846: 9844: 9840: 9834: 9831: 9829: 9826: 9824: 9821: 9819: 9816: 9814: 9811: 9809: 9806: 9805: 9803: 9801: 9797: 9791: 9788: 9786: 9783: 9779: 9776: 9774: 9771: 9770: 9769: 9766: 9765: 9762: 9757: 9749: 9744: 9742: 9737: 9735: 9730: 9729: 9726: 9714: 9711: 9709: 9706: 9704: 9701: 9699: 9698:Prime factors 9696: 9694: 9693:Prime numbers 9691: 9689: 9686: 9685: 9682: 9677: 9676:Number theory 9670: 9665: 9663: 9658: 9656: 9651: 9650: 9647: 9635: 9631: 9627: 9625: 9621: 9617: 9615: 9607: 9605: 9597: 9595: 9587: 9586: 9583: 9577: 9574: 9572: 9569: 9567: 9564: 9562: 9559: 9557: 9554: 9552: 9551:Modular forms 9549: 9547: 9544: 9543: 9541: 9537: 9531: 9528: 9526: 9523: 9521: 9518: 9516: 9513: 9510: 9506: 9503: 9501: 9498: 9496: 9493: 9491: 9488: 9486: 9483: 9481: 9478: 9476: 9475:Prime numbers 9473: 9471: 9468: 9466: 9463: 9461: 9458: 9456: 9453: 9452: 9450: 9446: 9440: 9437: 9435: 9432: 9429: 9425: 9421: 9418: 9415: 9411: 9408: 9405: 9401: 9397: 9394: 9392: 9389: 9387: 9384: 9382: 9379: 9376: 9372: 9368: 9364: 9361: 9358: 9357:Kummer theory 9354: 9350: 9346: 9342: 9338: 9335: 9334: 9332: 9328: 9324: 9323:Number theory 9317: 9312: 9310: 9305: 9303: 9298: 9297: 9294: 9288: 9285: 9283: 9280:entry in the 9279: 9278:Number Theory 9276: 9275: 9265: 9259: 9255: 9251: 9250: 9245: 9241: 9237: 9233: 9229: 9223: 9219: 9215: 9214: 9213:Number theory 9209: 9205: 9201: 9200: 9199: 9186: 9180: 9176: 9172: 9171: 9165: 9155: 9149: 9145: 9141: 9140: 9135: 9131: 9130:Ivan M. Niven 9127: 9126: 9125: 9123: 9114: 9110: 9106: 9096: 9090: 9086: 9082: 9081: 9076: 9072: 9071: 9070: 9060: 9058: 9054: 9050: 9049:Number theory 9046: 9041: 9040: 9028: 9022: 9018: 9017: 9012: 9008: 9004: 9000: 8996: 8992: 8978: 8974: 8970: 8966: 8962: 8957: 8952: 8948: 8944: 8937: 8932: 8922: 8916: 8912: 8908: 8907: 8901: 8891: 8885: 8881: 8877: 8873: 8869: 8865: 8862: 8856: 8850: 8846: 8842: 8838: 8834: 8830: 8828: 8822: 8816: 8808: 8807: 8802: 8798: 8794: 8792: 8789: 8786: 8783: 8778: 8777: 8772: 8771:Charles Henry 8768: 8764: 8763:Tannery, Paul 8760: 8756: 8751: 8747: 8741: 8737: 8733: 8732: 8727: 8723: 8712: 8708: 8707: 8702: 8698: 8694: 8687: 8683: 8679: 8675: 8671: 8664: 8660: 8656: 8652: 8648: 8644: 8640: 8636: 8632: 8627: 8616: 8612: 8611: 8605: 8601: 8595: 8591: 8590: 8585: 8581: 8577: 8573: 8569: 8565: 8561: 8557: 8553: 8549: 8545: 8541: 8529: 8525: 8521: 8517: 8513: 8509: 8505: 8501: 8497: 8493: 8487: 8483: 8479: 8475: 8461: 8457: 8453: 8449: 8445: 8438: 8434: 8430: 8426: 8420: 8412: 8406: 8402: 8401: 8396: 8391: 8381: 8375: 8371: 8370: 8365: 8361: 8357: 8346: 8342: 8338: 8328: 8322: 8318: 8314: 8313: 8307: 8303: 8299: 8295: 8291: 8286: 8276: 8270: 8266: 8265: 8260: 8259:Lam, Lay Yong 8256: 8245: 8241: 8240: 8235: 8231: 8227: 8223: 8217: 8213: 8209: 8205: 8193: 8189: 8185: 8180: 8176: 8170: 8166: 8162: 8157: 8147: 8143: 8142: 8137: 8133: 8129: 8125: 8121: 8115: 8111: 8107: 8106: 8101: 8097: 8093: 8087: 8083: 8079: 8075: 8064: 8060: 8059: 8054: 8050: 8046: 8036: 8030: 8026: 8022: 8021: 8016: 8012: 8008: 8004: 8000: 7990: 7984: 7980: 7976: 7972: 7968: 7954: 7947: 7943: 7939: 7935: 7929: 7925: 7924: 7919: 7915: 7911: 7905: 7901: 7896: 7891: 7886: 7882: 7878: 7874: 7869: 7858: 7857: 7852: 7848: 7844: 7838: 7834: 7830: 7829: 7823: 7819: 7815: 7811: 7807: 7803: 7799: 7795: 7791: 7787: 7781: 7777: 7773: 7769: 7765: 7761: 7750: 7749: 7744: 7740: 7730: 7726: 7725: 7719: 7716: 7711: 7705: 7701: 7697: 7696: 7691: 7687: 7683: 7679: 7675: 7671: 7670:Becker, Oskar 7667: 7662: 7658: 7654: 7650: 7645: 7642: 7636: 7632: 7627: 7617: 7611: 7607: 7603: 7599: 7598: 7593: 7589: 7586: 7580: 7576: 7571: 7570: 7556: 7552: 7548: 7544: 7539: 7534: 7530: 7526: 7522: 7518: 7514: 7507: 7493: 7489: 7483: 7479: 7475: 7471: 7467: 7460: 7445: 7441: 7435: 7431: 7427: 7424:. Routledge. 7423: 7422: 7414: 7406: 7402: 7398: 7394: 7390: 7386: 7381: 7376: 7372: 7368: 7361: 7353: 7347: 7332: 7328: 7324: 7320: 7314: 7310: 7309: 7301: 7286: 7282: 7276: 7272: 7268: 7264: 7263: 7256: 7250: 7246: 7240: 7231: 7225: 7221: 7217: 7212: 7205: 7201: 7198:Reprinted in 7195: 7191: 7187: 7181: 7177: 7173: 7169: 7165: 7161: 7157: 7153: 7152:Davis, Martin 7147: 7141:, p. 79. 7140: 7135: 7128: 7123: 7116: 7110: 7103: 7098: 7091: 7085: 7078: 7074: 7069: 7062: 7056: 7049: 7048:Goldfeld 2003 7044: 7037: 7031: 7024: 7019: 7012: 7007: 7000: 6995: 6988: 6982: 6975: 6970: 6963: 6958: 6956: 6948: 6944: 6938: 6931: 6927: 6921: 6915:, p. 14. 6914: 6909: 6902: 6897: 6890: 6885: 6878: 6873: 6857: 6853: 6849: 6843: 6836: 6831: 6829: 6821: 6816: 6809: 6804: 6797: 6792: 6785: 6780: 6773: 6768: 6761: 6756: 6749: 6744: 6737: 6733: 6728: 6721: 6716: 6709: 6705: 6700: 6693: 6689: 6684: 6677: 6672: 6665: 6660: 6653: 6648: 6641: 6636: 6629: 6624: 6617: 6612: 6606:, p. 92. 6605: 6600: 6593: 6588: 6581: 6576: 6569: 6565: 6561: 6557: 6553: 6549: 6544: 6537: 6533: 6528: 6513: 6509: 6503: 6499: 6498: 6490: 6484:, p. 54. 6483: 6479: 6475: 6470: 6463: 6459: 6455: 6450: 6443: 6438: 6431: 6427: 6422: 6415: 6410: 6403: 6399: 6395: 6390: 6383: 6379: 6375: 6371: 6366: 6359: 6354: 6347: 6342: 6335: 6330: 6323: 6318: 6311: 6307: 6303: 6299: 6295: 6291: 6285: 6278: 6273: 6266: 6260: 6253: 6248: 6246: 6238: 6233: 6226: 6221: 6212: 6203: 6188: 6184: 6178: 6172:, p. 82. 6171: 6166: 6160: 6155: 6149:, p. 310 6148: 6143: 6136: 6130: 6123: 6119: 6115: 6111: 6107: 6102: 6096:, p. 76. 6095: 6090: 6083: 6078: 6071: 6068: 6064: 6060: 6056: 6050: 6043: 6038: 6031: 6026: 6019: 6015: 6010: 6003: 5999: 5995: 5991: 5987: 5983: 5979: 5973: 5966: 5962: 5958: 5954: 5950: 5946: 5942: 5938: 5933: 5927:, p. 43. 5926: 5921: 5914: 5909: 5902: 5898: 5892: 5885: 5880: 5874:, p. 79) 5873: 5852: 5848: 5844: 5839: 5835: 5831: 5828: 5825: 5822: 5819: 5814: 5810: 5806: 5801: 5797: 5785: 5780: 5774:, p. 192 5773: 5769: 5765: 5760: 5753: 5748: 5744: 5731: 5727: 5711: 5708: 5703: 5695: 5692: 5689: 5686: 5680: 5675: 5667: 5664: 5661: 5658: 5647: 5643: 5627: 5624: 5619: 5615: 5611: 5606: 5602: 5591: 5584: 5580: 5576: 5572: 5568: 5564: 5560: 5556: 5553: − 5552: 5548: 5545: + 5544: 5540: 5536: 5533: + 5532: 5528: 5524: 5520: 5514: 5507: 5503: 5499: 5495: 5489: 5483: 5479: 5473: 5466: 5462: 5458: 5454: 5450: 5446: 5442: 5438: 5437:Royal Society 5434: 5430: 5426: 5422: 5416: 5409: 5405: 5401: 5397: 5379: 5371: 5368: 5365: 5362: 5354: 5350: 5346: 5342: 5337: 5333: 5329: 5325: 5321: 5317: 5313: 5309: 5305: 5302: − 5301: 5297: 5293: 5289: 5285: 5267: 5259: 5256: 5253: 5245: 5241: 5237: 5231: 5224: 5220: 5216: 5209: 5203: 5199: 5197: 5193: 5190: 5183: 5179: 5173: 5166: 5164: 5160: 5157: 5149: 5145: 5140: 5134: 5132: 5125: 5123: 5115: 5111: 5107: 5103: 5098: 5091: 5087: 5083: 5082: 5077: 5073: 5069: 5065: 5059: 5053:, p. 13) 5052: 5048: 5042: 5038: 5029: 5026: 5024: 5023:p-adic number 5021: 5019: 5016: 5014: 5011: 5010: 5006: 5000: 4995: 4988: 4986: 4985: 4980: 4979: 4974: 4961: 4957: 4953: 4950: 4949:modular forms 4946: 4943: 4940: 4937: 4933: 4930: 4927: 4923: 4920: 4917: 4914: 4913: 4912: 4909: 4907: 4903: 4899: 4894: 4890: 4880: 4877: 4873: 4869: 4864: 4862: 4858: 4853: 4851: 4845: 4843: 4839: 4835: 4819: 4816: 4813: 4810: 4807: 4804: 4801: 4793: 4789: 4784: 4782: 4778: 4771: 4767: 4764:used to find 4763: 4759: 4754: 4749: 4739: 4725: 4705: 4685: 4682: 4679: 4671: 4667: 4651: 4643: 4639: 4635: 4631: 4627: 4611: 4603: 4602: 4597: 4592: 4578: 4558: 4555: 4552: 4549: 4546: 4543: 4540: 4537: 4534: 4531: 4528: 4525: 4522: 4519: 4516: 4513: 4510: 4507: 4504: 4501: 4481: 4461: 4454: 4448: 4444: 4434: 4432: 4428: 4423: 4408: 4400: 4395: 4393: 4387: 4377: 4375: 4371: 4370:computability 4360: 4358: 4354: 4350: 4346: 4342: 4337: 4335: 4331: 4325: 4321: 4317: 4301: 4281: 4261: 4237: 4233: 4229: 4224: 4216: 4212: 4208: 4205: 4202: 4177: 4157: 4154: 4148: 4145: 4142: 4116: 4112: 4108: 4088: 4080: 4075: 4061: 4058: 4052: 4049: 4046: 4040: 4032: 4015: 4012: 4006: 4003: 4000: 3994: 3974: 3971: 3965: 3962: 3959: 3953: 3945: 3941: 3925: 3905: 3902: 3896: 3893: 3890: 3884: 3871: 3867: 3862: 3858: 3844: 3841: 3836: 3832: 3828: 3823: 3819: 3798: 3794: 3790: 3787: 3784: 3764: 3760: 3756: 3753: 3750: 3728: 3724: 3720: 3715: 3711: 3707: 3702: 3698: 3689: 3685: 3666: 3663: 3660: 3637: 3634: 3631: 3626: 3622: 3618: 3613: 3609: 3601: 3596: 3594: 3590: 3586: 3582: 3578: 3573: 3571: 3567: 3561: 3551: 3549: 3545: 3540: 3538: 3534: 3530: 3526: 3525:abelian group 3522: 3518: 3514: 3510: 3506: 3502: 3498: 3494: 3490: 3486: 3482: 3478: 3474: 3470: 3466: 3462: 3457: 3455: 3451: 3433: 3430: 3425: 3422: 3400: 3397: 3392: 3389: 3369: 3349: 3324: 3321: 3316: 3313: 3302: 3299: 3294: 3291: 3285: 3282: 3262: 3259: 3256: 3253: 3250: 3230: 3223:, the number 3208: 3205: 3195: 3191: 3187: 3186:ideal numbers 3182: 3180: 3164: 3144: 3124: 3102: 3097: 3094: 3091: 3083: 3079: 3075: 3071: 3066: 3064: 3063: 3062:number fields 3059:, or shortly 3058: 3057: 3040: 3037: 3034: 3031: 3026: 3022: 3018: 3015: 3010: 3006: 2999: 2995: 2991: 2985: 2980: 2976: 2955: 2935: 2932: 2926: 2920: 2912: 2908: 2902: 2892: 2890: 2886: 2882: 2878: 2874: 2873:prime numbers 2870: 2866: 2861: 2859: 2855: 2854:modular forms 2851: 2847: 2846:sieve methods 2843: 2842:circle method 2839: 2835: 2831: 2827: 2823: 2819: 2814: 2812: 2804: 2801: 2797: 2793: 2792: 2791: 2789: 2782: 2778: 2774: 2773:modular group 2769: 2762: 2758: 2754: 2750: 2749:complex plane 2746: 2742: 2738: 2733: 2723: 2716: 2712: 2707: 2703: 2701: 2697: 2693: 2689: 2685: 2681: 2677: 2673: 2669: 2668: 2652: 2648: 2646: 2645:modular forms 2642: 2638: 2637:zeta function 2634: 2630: 2625: 2621: 2617: 2613: 2609: 2601: 2597: 2594: 2593:Galois theory 2590: 2586: 2582: 2579: 2575: 2574: 2573: 2568: 2564: 2558: 2554: 2545: 2543: 2539: 2533: 2529: 2527: 2523: 2519: 2515: 2511: 2502: 2498: 2496: 2492: 2476: 2473: 2470: 2450: 2447: 2442: 2438: 2434: 2431: 2426: 2422: 2418: 2415: 2410: 2406: 2402: 2394: 2390: 2386: 2382: 2366: 2362: 2358: 2355: 2350: 2346: 2342: 2334: 2330: 2326: 2318: 2306: 2302: 2298: 2293: 2286: 2282: 2279: 2276: 2258: 2254: 2250: 2247: 2242: 2238: 2229: 2226: 2223: 2219: 2215: 2211: 2207: 2203: 2199: 2196: 2193: 2192: 2188: 2185: 2181: 2163: 2159: 2155: 2150: 2146: 2142: 2137: 2133: 2124: 2106: 2098: 2095: 2092: 2070: 2066: 2062: 2057: 2053: 2049: 2046: 2038: 2034: 2031: 2030: 2029: 2027: 2023: 2014: 1990: 1987: 1984: 1962: 1958: 1954: 1949: 1945: 1941: 1936: 1932: 1923: 1919: 1902: 1898: 1894: 1889: 1885: 1881: 1876: 1872: 1849: 1845: 1841: 1836: 1832: 1828: 1823: 1819: 1810: 1806: 1790: 1787: 1782: 1778: 1774: 1771: 1766: 1762: 1753: 1750: 1732: 1728: 1724: 1719: 1715: 1692: 1688: 1684: 1679: 1675: 1666: 1662: 1658: 1654: 1640: 1635: 1627: 1624: 1619: 1616: 1613: 1609: 1600: 1596: 1592: 1589: 1586: 1583: 1579: 1575: 1571: 1567: 1566: 1565: 1562: 1560: 1554: 1550: 1536: 1534: 1530: 1526: 1516: 1514: 1510: 1506: 1502: 1498: 1494: 1490: 1486: 1482: 1474: 1473: 1468: 1467: 1466:Selenographia 1462: 1458: 1453: 1449: 1439: 1437: 1432: 1430: 1426: 1422: 1417: 1413: 1412:Pell equation 1408: 1406: 1402: 1384: 1378: 1367: 1363: 1359: 1356: 1334: 1328: 1317: 1313: 1309: 1306: 1297: 1295: 1285: 1283: 1279: 1278: 1259: 1255: 1251: 1245: 1242: 1239: 1236: 1233: 1227: 1205: 1201: 1197: 1191: 1188: 1185: 1179: 1171: 1167: 1166: 1161: 1154: 1150: 1146: 1142: 1137: 1128: 1126: 1125:Indian school 1122: 1118: 1114: 1110: 1106: 1103:published an 1102: 1097: 1095: 1091: 1087: 1083: 1079: 1075: 1070: 1068: 1064: 1061:. (Book X of 1060: 1056: 1052: 1034: 1029: 1026: 1023: 1018: 1013: 1008: 998: 994: 993: 988: 987:number theory 984: 980: 975: 973: 967: 963: 961: 957: 953: 950: 948: 944: 943: 938: 932: 922: 919: 917: 913: 912: 907: 897: 893: 889: 885: 881: 880: 875: 870: 868: 864: 860: 856: 855:cubic numbers 852: 848: 844: 839: 837: 833: 829: 828: 823: 819: 801: 791: 787: 783: 768: 757: 753: 748: 746: 742: 738: 734: 730: 725: 723: 718: 704: 700: 696: 688: 669: 664: 659: 654: 648: 645: 640: 637: 633: 627: 624: 618: 613: 610: 607: 602: 597: 592: 586: 583: 578: 575: 571: 565: 562: 556: 547: 546: 545: 544: 539: 537: 533: 515: 511: 507: 502: 498: 494: 489: 485: 461: 458: 455: 452: 449: 438: 434: 430: 421: 402: 400: 396: 392: 391:number theory 388: 384: 383: 378: 374: 373: 368: 364: 360: 359: 354: 353:number theory 350: 345: 343: 339: 335: 331: 327: 323: 318: 316: 312: 308: 304: 303:prime numbers 300: 296: 292: 288: 284: 280: 279: 274: 273:Number theory 263: 258: 256: 251: 249: 244: 243: 241: 240: 237: 233: 229: 228: 217: 214: 212: 209: 207: 204: 202: 199: 197: 194: 192: 189: 187: 184: 182: 179: 177: 174: 173: 167: 166: 159: 155: 152: 150: 147: 145: 141: 138: 136: 133: 131: 127: 124: 122: 119: 117: 114: 112: 111:Number theory 109: 108: 105: 100: 99: 96: 95: 90: 87: 85: 82: 81: 80: 79: 76: 73: 72: 68: 67: 61: 57: 53: 52:prime numbers 48: 42: 35: 34: 19: 11272:Cyberwarfare 10931:Cryptography 10324: 10312: 10300: 10281: 10214:Optimization 10076:Differential 10024: 10000:Differential 9967:Order theory 9962:Graph theory 9866:Group theory 9675: 9448:Key concepts 9375:sieve theory 9322: 9248: 9212: 9197: 9188:. Retrieved 9169: 9157:. Retrieved 9138: 9122:Apostol 1981 9119: 9112: 9098:. Retrieved 9079: 9068: 9042: 9030:. Retrieved 9015: 8998: 8984:. Retrieved 8946: 8942: 8924:. Retrieved 8905: 8893:. Retrieved 8875: 8860: 8840: 8805: 8775: 8754: 8730: 8715:. Retrieved 8705: 8686:the original 8673: 8669: 8634: 8630: 8619:. Retrieved 8609: 8588: 8584:Plofker, Kim 8575: 8571: 8555: 8551: 8532:. Retrieved 8507: 8481: 8467:. Retrieved 8447: 8443: 8399: 8383:. Retrieved 8368: 8348:. Retrieved 8341:Milne, J. S. 8330:. Retrieved 8311: 8289: 8278:. Retrieved 8263: 8248:. Retrieved 8238: 8211: 8196:. Retrieved 8187: 8184:"Pythagoras" 8160: 8149:. Retrieved 8140: 8104: 8081: 8067:. Retrieved 8057: 8038:. Retrieved 8019: 8015:Leader, Imre 7992:. Retrieved 7978: 7960:. Retrieved 7926:. Springer. 7922: 7899: 7880: 7876: 7861:. Retrieved 7855: 7827: 7801: 7797: 7771: 7753:. Retrieved 7747: 7732:. Retrieved 7723: 7715:1968 edition 7694: 7677: 7673: 7648: 7630: 7619:. Retrieved 7596: 7574: 7523:(1): 51–58. 7520: 7516: 7506: 7496:, retrieved 7469: 7459: 7448:. Retrieved 7420: 7413: 7370: 7366: 7360: 7335:. Retrieved 7307: 7300: 7289:. Retrieved 7261: 7255: 7239: 7230: 7215: 7211: 7199: 7167: 7146: 7139:Edwards 2000 7134: 7129:, p. 2. 7122: 7109: 7097: 7084: 7076: 7068: 7063:, p. 1. 7055: 7043: 7030: 7018: 7006: 7001:, p. 1. 6994: 6981: 6976:, p. 1. 6969: 6964:, p. 7. 6962:Apostol 1976 6937: 6932:, p. 16 6925: 6920: 6908: 6896: 6884: 6872: 6860:. Retrieved 6842: 6815: 6803: 6796:Edwards 1983 6791: 6779: 6767: 6755: 6743: 6727: 6715: 6699: 6694:, p. 37 6683: 6671: 6666:, p. 9. 6659: 6647: 6635: 6623: 6611: 6599: 6587: 6575: 6563: 6555: 6543: 6538:, p. 56 6527: 6516:. Retrieved 6496: 6489: 6482:Mahoney 1994 6474:Mahoney 1994 6469: 6458:Mahoney 1994 6449: 6437: 6421: 6409: 6402:Plofker 2008 6398:Pingree 1970 6394:Pingree 1968 6389: 6374:Hopkins 1990 6365: 6353: 6346:Plofker 2008 6341: 6334:Plofker 2008 6329: 6322:Mumford 2010 6317: 6301: 6290:Plofker 2008 6284: 6277:Mumford 2010 6272: 6265:Plofker 2008 6259: 6252:Plofker 2008 6232: 6220: 6211: 6202: 6191:. Retrieved 6177: 6165: 6154: 6142: 6129: 6113: 6105: 6101: 6089: 6077: 6066: 6061:), cited in 6054: 6049: 6037: 6025: 6009: 5997: 5990:O'Grady 2004 5982:Huffman 2011 5977: 5972: 5961:Guthrie 1987 5956: 5945:Guthrie 1987 5940: 5932: 5920: 5913:Friberg 1981 5908: 5895:Neugebauer ( 5891: 5879: 5779: 5767: 5759: 5754:, p. 1. 5747: 5725: 5645: 5641: 5590: 5578: 5574: 5570: 5566: 5562: 5554: 5550: 5546: 5542: 5538: 5534: 5530: 5526: 5523:isomorphisms 5518: 5513: 5498:large sieves 5488: 5472: 5456: 5443:, 1666) and 5424: 5415: 5395: 5352: 5348: 5344: 5340: 5315: 5311: 5307: 5303: 5299: 5295: 5291: 5287: 5283: 5243: 5239: 5235: 5230: 5208: 5201: 5195: 5194: 5188: 5186: 5177: 5172: 5162: 5161: 5155: 5152: 5143: 5139: 5127: 5118: 5106:Plimpton 322 5097: 5085: 5079: 5071: 5058: 5041: 5018:Finite field 4984:Fermat Prize 4982: 4976: 4970: 4916:Cryptography 4910: 4900:courses for 4893:Donald Knuth 4886: 4883:Applications 4865: 4854: 4846: 4841: 4791: 4785: 4781:al-Khwārizmī 4776: 4774: 4758:Lehmer sieve 4641: 4638:model theory 4625: 4599: 4595: 4593: 4453:infinite set 4450: 4424: 4396: 4389: 4366: 4352: 4348: 4344: 4338: 4076: 4030: 3943: 3939: 3875: 3687: 3683: 3597: 3592: 3588: 3584: 3574: 3565: 3563: 3541: 3520: 3516: 3512: 3508: 3505:Galois group 3500: 3496: 3492: 3488: 3484: 3480: 3476: 3472: 3468: 3464: 3460: 3458: 3193: 3189: 3185: 3183: 3081: 3069: 3067: 3060: 3054: 2906: 2904: 2869:prime ideals 2862: 2815: 2811:sieve theory 2808: 2802:analysis; or 2787: 2786: 2756: 2752: 2744: 2720: 2699: 2665: 2663: 2649: 2628: 2605: 2589:group theory 2577: 2571: 2557:Ernst Kummer 2540:'s work and 2535: 2531: 2521: 2509: 2507: 2383: 2323: 2297:Andrew Wiles 2280: 2227: 2218:power series 2210:distribution 2197: 2189: 2183: 2179: 2032: 2019: 1808: 1660: 1656: 1598: 1594: 1573: 1563: 1557: 1532: 1522: 1500: 1492: 1484: 1478: 1470: 1464: 1433: 1409: 1404: 1400: 1298: 1294:trigonometry 1291: 1275: 1169: 1163: 1157: 1144: 1113:Eratosthenes 1098: 1093: 1085: 1077: 1073: 1071: 991: 986: 982: 978: 976: 971: 969: 965: 954: 951: 946: 940: 934: 920: 909: 895: 877: 871: 840: 835: 831: 825: 749: 726: 719: 684: 540: 535: 429:Plimpton 322 426: 398: 395:arithmetical 394: 390: 386: 380: 370: 356: 355:. (The word 352: 348: 346: 338:real numbers 319: 282: 276: 272: 271: 110: 60:independence 32: 11282:Video games 11262:Digital art 11019:Concurrency 10888:Data mining 10800:Probability 10540:Interpreter 10326:WikiProject 10169:Game theory 10149:Probability 9886:Homological 9876:Multilinear 9856:Commutative 9833:Type theory 9800:Foundations 9756:mathematics 9634:Wikiversity 9556:L-functions 9045:Citizendium 9011:Weil, André 8564:Pingree, D. 7538:10261/18003 7038:, p. 1 6989:, section 1 6568:Fermat 1679 6556:Varia Opera 6414:Rashed 1980 6298:Brahmagupta 6147:Dauben 2007 6059:Jowett 1871 6030:Becker 1936 6014:Becker 1936 6002:Morrow 1992 5986:Morrow 1992 5947:) cited in 5784:Robson 2001 5772:Robson 2001 5640:. We allow 5246:, we write 5131:Robson 2001 5122:Robson 2001 5114:Robson 2001 5110:Robson 2001 5102:Robson 2001 5066:. In 1952, 5047:T. L. Heath 4975:awards the 4392:independent 4351:. The term 3467:of a field 2850:L-functions 2715:Terence Tao 2624:L-functions 2518:congruences 1593:(1640): if 1533:Arithmetica 1529:Renaissance 1493:Arithmetica 1429:Bhāskara II 1414:, in which 1170:Arithmetica 1165:Arithmetica 1145:Arithmetica 906:Qin Jiushao 836:proportions 532:brute force 201:Linguistics 191:Computation 186:Geosciences 149:Probability 75:Mathematics 56:Ulam spiral 11340:Glossaries 11212:E-commerce 10805:Statistics 10748:Algorithms 10545:Middleware 10401:Peripheral 10154:Statistics 10033:Arithmetic 9995:Arithmetic 9861:Elementary 9828:Set theory 9515:Arithmetic 9190:2016-02-28 9159:2016-02-28 9100:2016-03-02 9075:G.H. Hardy 9032:2016-02-28 8986:2012-04-08 8926:2016-02-28 8895:2016-02-28 8797:Iamblichus 8717:2016-02-28 8621:2016-02-28 8534:7 February 8469:2021-04-28 8450:(3): 387. 8385:2016-02-28 8332:2016-02-28 8280:2016-02-28 8250:2012-04-10 8239:Theaetetus 8198:7 February 8151:2016-02-28 8144:. Oxford: 8069:2012-04-10 8040:2016-02-28 7994:2016-02-28 7962:2016-02-28 7863:2016-02-28 7755:2016-02-28 7734:2016-02-28 7680:: 533–553. 7621:2016-02-28 7498:2023-02-22 7450:2023-02-22 7337:2023-02-22 7327:1004350753 7311:. London. 7291:2023-02-22 7194:0346.02026 7127:Milne 2017 6518:2019-08-06 6382:Smith 1958 6310:Clark 1930 6294:Clark 1930 6225:Vardi 1998 6193:2017-02-20 6116:(based on 6114:Suan Ching 6094:Heath 1921 6055:Theaetetus 5937:Iamblichus 5739:References 5494:Brun sieve 5355:such that 5215:Nicomachus 5064:Serre 1996 5051:Heath 1921 3682:such that 3537:Eisenstein 2711:Paul Erdős 2700:elementary 2667:elementary 2208:, and the 2202:partitions 1461:Al-Haytham 1421:chakravala 1416:Archimedes 1405:pulveriser 1131:Diophantus 1109:Archimedes 1051:Theaetetus 992:Theaetetus 983:arithmetic 979:arithmetic 960:Pythagoras 896:Da-yan-shu 786:irrational 756:Pythagoras 737:Pythagoras 477:such that 387:arithmetic 358:arithmetic 349:arithmetic 326:analytical 278:arithmetic 211:Philosophy 154:Statistics 144:Set theory 41:Numerology 11161:Rendering 11156:Animation 10787:computing 10738:Semantics 10436:Processor 10081:Geometric 10071:Algebraic 10010:Euclidean 9985:Algebraic 9881:Universal 9246:(1996) . 9136:(2008) . 9111:(2003) . 9047:article " 8951:CiteSeerX 8815:cite book 8728:(1996) . 8651:120885025 8568:al-Fazari 8456:1088-9477 8419:cite book 7692:(1991) . 7547:0929-8215 7405:119290777 7380:1101.3116 7346:cite book 6947:Weil 1984 6901:Weil 1984 6889:Weil 1984 6877:Weil 1984 6835:Weil 1984 6820:Weil 1984 6784:Weil 1984 6748:Weil 1984 6732:Weil 1984 6720:Weil 1984 6708:Weil 1984 6688:Weil 1984 6676:Weil 1984 6652:Weil 1984 6640:Weil 1984 6628:Weil 1984 6604:Weil 1984 6560:Weil 1984 6552:Weil 1984 6536:Weil 1984 6462:Weil 1984 6454:Weil 1984 6442:Weil 1984 6380:cited in 6237:Weil 1984 6118:Qian 1963 5807:− 5752:Long 1972 5449:Weil 1984 5433:Weil 1984 5429:Weil 1984 5421:Weil 1984 5408:Weil 1984 5369:≡ 5330:that the 5257:≡ 5225:, Ch. IV. 5068:Davenport 4838:Āryabhaṭa 4777:algorithm 4544:… 4427:heuristic 4206:− 3533:Kronecker 3475:contains 3465:extension 3431:− 3426:− 3398:− 3342:; all of 3322:− 3317:− 3300:− 3260:⋅ 3206:− 3016:− 2828:, or the 2747:) in the 2664:The term 2616:valuation 2608:cyclotomy 2301:his proof 2096:≡ 1988:≥ 1772:− 1625:≡ 1617:− 1525:Fibonacci 1505:al-Karajī 1501:al-Fakhri 1481:Al-Ma'mun 1360:≡ 1310:≡ 1099:In 1773, 1027:… 997:Theodorus 884:Āryabhaṭa 843:polygonal 818:Theodorus 579:− 216:Education 206:Economics 181:Chemistry 11355:Category 11320:Category 11148:Graphics 10923:Security 10592:Compiler 10491:Networks 10388:Hardware 10302:Category 10058:Topology 10005:Discrete 9990:Analytic 9977:Geometry 9949:Discrete 9904:Calculus 9896:Analysis 9851:Abstract 9790:Glossary 9773:Timeline 9703:Divisors 9624:Wikibook 9594:Category 9210:(1966). 9013:(1984). 8977:Archived 8870:(2007). 8835:(1984). 8803:(1818). 8788:Volume 3 8785:Volume 2 8782:Volume 1 8769:(1891). 8736:Springer 8711:Archived 8703:(1888). 8661:(2001). 8615:Archived 8586:(2008). 8528:Archived 8480:(1969). 8460:Archived 8397:(1992). 8366:(2007). 8302:77171950 8244:Archived 8236:(1871). 8192:Archived 8138:(1921). 8080:(1987). 8063:Archived 8055:(1920). 8049:Porphyry 8017:(eds.). 8005:(2008). 7953:Archived 7944:(2003). 7853:(1679). 7776:Springer 7770:(2000). 7745:(1817). 7606:Springer 7594:(1976). 7555:12232457 7492:archived 7444:Archived 7331:Archived 7285:Archived 6862:16 March 6856:Archived 6512:Archived 6430:Xylander 6187:Archived 6067:See also 5953:Porphyry 5768:takiltum 5583:solvable 5527:Gal(C/R) 5502:harmonic 5425:practice 5298:divides 4991:See also 4792:Elements 4254:, where 3918:, where 3117:, where 2889:sequence 2836:and the 2824:(or the 2761:argument 2600:analytic 2026:Goldbach 1977:for all 1582:divisors 1576:IX) and 1574:Elements 1485:Sindhind 1425:Jayadeva 1094:Elements 1086:Elements 1078:Elements 1074:Elements 956:Eusebius 908:'s 1247 849:. While 822:Hippasus 543:identity 536:takiltum 379:, as in 369:, as in 291:integers 130:Analysis 126:Calculus 116:Geometry 11330:Outline 10314:Commons 10096:Applied 10066:General 9843:Algebra 9768:History 9455:Numbers 9236:0195803 8973:2589706 8773:(ed.). 8395:Proclus 8350:7 April 8128:2445243 7818:2690368 7661:0568909 7566:Sources 7385:Bibcode 7166:(ed.). 7117:, p. 1. 6053:Plato, 5978:Busiris 5457:amateur 5290:modulo 5282:(read " 5221:". See 5191:: Male. 4932:Physics 4842:kuṭṭaka 3598:In the 3581:surface 2909:is any 2832:), the 2800:complex 2775:on the 2684:Selberg 2629:complex 2508:In his 1667:, then 1665:coprime 1601:, then 1401:kuṭṭaka 1105:epigram 1101:Lessing 888:Kuṭṭaka 832:lengths 827:numbers 410:Origins 405:History 196:Biology 176:Physics 121:Algebra 84:History 10015:Finite 9871:Linear 9778:Future 9754:Major 9678:tables 9330:Fields 9260: 9234: 9224: 9181: 9150: 9091: 9023: 8971: 8953: 8917: 8886: 8851: 8742: 8649: 8596: 8488: 8454: 8407: 8376: 8323: 8300: 8271: 8218: 8171: 8126: 8116: 8088: 8031: 7985: 7930: 7906: 7839: 7816: 7782: 7706: 7659: 7637: 7612: 7581: 7553: 7545: 7484: 7436: 7403: 7325: 7315: 7277: 7247: 7222: 7192: 7182: 6504: 6432:, 1575 6426:Bachet 5445:Russia 5404:Bachet 5196:Method 5189:Answer 5163:Method 5156:Answer 4967:Prizes 4960:octave 4934:: The 4924:: The 4766:primes 4666:groups 4316:height 4129:(with 3546:. The 3523:is an 3450:Kummer 3188:, the 3074:ideals 2820:, the 2299:about 1544:Fermat 1067:Pappus 947:Euclid 890:– see 741:Thales 375:, and 10733:Logic 10574:tools 10242:lists 9785:Lists 9758:areas 9688:Bases 9470:Unity 8980:(PDF) 8969:JSTOR 8939:(PDF) 8689:(PDF) 8666:(PDF) 8647:S2CID 8463:(PDF) 8440:(PDF) 8230:Plato 8009:. In 7956:(PDF) 7949:(PDF) 7814:JSTOR 7700:Wiley 7551:S2CID 7401:S2CID 7375:arXiv 5730:torus 5400:above 5336:order 5332:order 5320:Gauss 5158:: 23. 5076:Hardy 5034:Notes 4670:rings 4031:genus 3944:genus 3940:genus 3870:torus 3577:curve 3519:over 3515:) of 3078:norms 2680:Erdős 2008:Euler 1403:, or 1149:Latin 942:Plato 904:) in 892:below 745:Egypt 140:Logic 104:Areas 89:Index 10572:and 10445:Form 10441:Size 9258:ISBN 9222:ISBN 9179:ISBN 9148:ISBN 9089:ISBN 9057:GFDL 9021:ISBN 8915:ISBN 8884:ISBN 8849:ISBN 8821:link 8740:ISBN 8594:ISBN 8536:2012 8514:etc. 8486:ISBN 8452:ISSN 8425:link 8405:ISBN 8374:ISBN 8352:2020 8321:ISBN 8298:LCCN 8269:ISBN 8216:ISBN 8200:2012 8169:ISBN 8114:ISBN 8086:ISBN 8029:ISBN 7983:ISBN 7928:ISBN 7904:ISBN 7837:ISBN 7780:ISBN 7704:ISBN 7635:ISBN 7610:ISBN 7579:ISBN 7543:ISSN 7482:ISBN 7434:ISBN 7352:link 7323:OCLC 7313:ISBN 7275:ISBN 7245:ISBN 7220:ISBN 7180:ISBN 6864:2016 6852:WGBH 6502:ISBN 5644:and 5504:and 5328:fact 5310:and 5238:and 4971:The 4698:and 4445:and 4332:and 4225:< 3686:and 3579:, a 3535:and 3507:Gal( 3415:and 3275:and 3192:and 3137:and 3076:and 2848:and 2798:and 2796:real 2713:and 2682:and 2391:and 1663:are 1659:and 1503:(by 1450:and 945:and 872:The 834:and 389:for 293:and 275:(or 156:and 142:and 128:and 8961:doi 8947:105 8678:doi 8639:doi 7885:doi 7806:doi 7533:hdl 7525:doi 7474:doi 7426:doi 7393:doi 7267:doi 7190:Zbl 5549:to 5537:of 5519:L/K 5480:or 5376:mod 5322:'s 5264:mod 4861:RSA 4190:if 4137:gcd 4033:of 3499:of 3471:if 2968:of 2905:An 2647:). 2497:). 2303:of 2184:n=3 2180:n=4 2103:mod 1655:If 1632:mod 1472:sic 1375:mod 1325:mod 1220:or 1151:by 1143:'s 1111:to 985:or 901:大衍術 886:'s 845:or 784:is 344:). 317:). 281:or 11357:: 10443:/ 9426:, 9402:, 9373:, 9369:, 9355:, 9351:, 9347:, 9343:, 9252:. 9232:MR 9230:. 9220:. 9206:; 9177:. 9146:. 9087:. 8975:. 8967:. 8959:. 8945:. 8941:. 8913:. 8909:. 8882:. 8847:. 8817:}} 8813:{{ 8799:; 8765:; 8738:. 8699:; 8674:28 8672:. 8668:. 8645:. 8635:22 8633:. 8576:28 8574:. 8566:; 8556:26 8554:. 8546:; 8502:; 8458:. 8448:57 8446:. 8442:. 8421:}} 8417:{{ 8362:; 8319:. 8296:. 8242:. 8232:; 8167:. 8124:MR 8122:. 8112:. 8051:; 8027:. 8023:. 7951:. 7879:. 7875:. 7835:. 7812:. 7802:56 7800:. 7778:. 7766:; 7727:. 7702:. 7688:; 7657:MR 7655:. 7651:. 7608:. 7604:. 7600:. 7549:. 7541:. 7531:. 7521:31 7519:. 7515:. 7490:, 7480:, 7468:, 7442:. 7432:. 7399:. 7391:. 7383:. 7371:83 7369:. 7348:}} 7344:{{ 7329:. 7321:. 7283:. 7273:. 7202:, 7188:. 7170:. 7158:; 7154:; 7079:." 6954:^ 6850:. 6827:^ 6570:). 6510:. 6300:, 6244:^ 6185:. 5955:, 5939:, 5555:iy 5547:iy 5535:iy 5074:. 4987:. 4908:. 4756:A 4636:, 4632:, 4591:? 4556:10 4494:, 4433:. 3777:, 3456:. 3382:, 3362:, 2992:11 2844:, 2743:ζ( 2591:, 2587:, 2544:. 2204:, 1535:. 1515:. 1438:. 1349:, 1280:a 1084:; 1035:17 962:: 918:. 861:, 853:, 788:. 747:. 401:. 10381:. 10361:e 10354:t 10347:v 9747:e 9740:t 9733:v 9668:e 9661:t 9654:v 9511:) 9507:( 9460:0 9430:) 9422:( 9416:) 9412:( 9406:) 9398:( 9377:) 9365:( 9359:) 9339:( 9315:e 9308:t 9301:v 9266:. 9238:. 9193:. 9162:. 9103:. 9059:. 9035:. 9005:. 8989:. 8963:: 8929:. 8898:. 8857:. 8823:) 8748:. 8720:. 8680:: 8653:. 8641:: 8624:. 8602:. 8578:. 8558:. 8538:. 8494:. 8472:. 8427:) 8413:. 8388:. 8354:. 8335:. 8304:. 8283:. 8253:. 8224:. 8202:. 8177:. 8154:. 8130:. 8094:. 8072:. 8043:. 7997:. 7965:. 7936:. 7912:. 7893:. 7887:: 7881:8 7866:. 7845:. 7820:. 7808:: 7788:. 7758:. 7737:. 7712:. 7678:3 7663:. 7624:. 7557:. 7535:: 7527:: 7476:: 7453:. 7428:: 7407:. 7395:: 7387:: 7377:: 7354:) 7340:. 7294:. 7269:: 7196:. 7050:. 6866:. 6521:. 6360:. 6196:. 6084:. 6072:. 6032:. 5858:) 5853:2 5849:q 5845:+ 5840:2 5836:p 5832:, 5829:q 5826:p 5823:2 5820:, 5815:2 5811:q 5802:2 5798:p 5794:( 5726:a 5712:7 5709:+ 5704:3 5700:) 5696:i 5693:d 5690:+ 5687:c 5684:( 5681:= 5676:2 5672:) 5668:i 5665:b 5662:+ 5659:a 5656:( 5646:y 5642:x 5628:7 5625:+ 5620:3 5616:x 5612:= 5607:2 5603:y 5579:x 5577:( 5575:f 5571:x 5567:x 5565:( 5563:f 5551:x 5543:x 5539:C 5531:x 5508:. 5406:( 5396:p 5380:p 5372:1 5366:a 5363:x 5353:x 5349:p 5345:a 5341:p 5316:m 5312:b 5308:a 5304:b 5300:a 5296:m 5292:m 5288:b 5284:a 5268:m 5260:b 5254:a 5244:m 5240:b 5236:a 5120:( 5092:) 4820:c 4817:= 4814:y 4811:b 4808:+ 4805:x 4802:a 4726:A 4718:· 4706:A 4686:A 4683:+ 4680:A 4652:A 4612:A 4579:A 4559:b 4553:+ 4550:a 4547:, 4541:, 4538:b 4535:3 4532:+ 4529:a 4526:, 4523:b 4520:2 4517:+ 4514:a 4511:, 4508:b 4505:+ 4502:a 4482:a 4462:A 4409:0 4334:e 4329:π 4302:x 4282:x 4262:c 4238:c 4234:q 4230:1 4221:| 4217:q 4213:/ 4209:a 4203:x 4199:| 4178:x 4158:1 4155:= 4152:) 4149:q 4146:, 4143:a 4140:( 4117:q 4113:/ 4109:a 4089:x 4062:0 4059:= 4056:) 4053:y 4050:, 4047:x 4044:( 4041:f 4016:0 4013:= 4010:) 4007:y 4004:, 4001:x 3998:( 3995:f 3975:0 3972:= 3969:) 3966:y 3963:, 3960:x 3957:( 3954:f 3926:f 3906:0 3903:= 3900:) 3897:y 3894:, 3891:x 3888:( 3885:f 3845:1 3842:= 3837:2 3833:y 3829:+ 3824:2 3820:x 3799:c 3795:/ 3791:b 3788:= 3785:y 3765:c 3761:/ 3757:a 3754:= 3751:x 3729:2 3725:c 3721:= 3716:2 3712:b 3708:+ 3703:2 3699:a 3688:y 3684:x 3670:) 3667:y 3664:, 3661:x 3658:( 3638:, 3635:1 3632:= 3627:2 3623:y 3619:+ 3614:2 3610:x 3585:n 3521:K 3517:L 3513:K 3511:/ 3509:L 3501:K 3497:L 3493:Q 3489:R 3485:R 3481:C 3477:K 3473:L 3469:K 3461:L 3434:5 3423:1 3401:5 3393:+ 3390:1 3370:3 3350:2 3330:) 3325:5 3314:1 3311:( 3308:) 3303:5 3295:+ 3292:1 3289:( 3286:= 3283:6 3263:3 3257:2 3254:= 3251:6 3231:6 3209:5 3165:d 3145:b 3125:a 3103:d 3098:b 3095:+ 3092:a 3041:0 3038:= 3035:9 3032:+ 3027:2 3023:x 3019:7 3011:3 3007:x 3003:) 3000:2 2996:/ 2989:( 2986:+ 2981:5 2977:x 2956:x 2936:0 2933:= 2930:) 2927:x 2924:( 2921:f 2783:. 2763:. 2757:s 2753:s 2745:s 2477:5 2474:= 2471:n 2451:0 2448:= 2443:2 2439:z 2435:c 2432:+ 2427:2 2423:y 2419:b 2416:+ 2411:2 2407:x 2403:a 2367:2 2363:Y 2359:n 2356:+ 2351:2 2347:X 2343:m 2307:. 2277:. 2259:2 2255:y 2251:N 2248:+ 2243:2 2239:x 2224:. 2164:2 2160:z 2156:= 2151:4 2147:y 2143:+ 2138:4 2134:x 2107:4 2099:1 2093:p 2071:2 2067:y 2063:+ 2058:2 2054:x 2050:= 2047:p 1991:3 1985:n 1963:n 1959:z 1955:= 1950:n 1946:y 1942:+ 1937:n 1933:x 1903:3 1899:z 1895:= 1890:3 1886:y 1882:+ 1877:3 1873:x 1850:4 1846:z 1842:= 1837:4 1833:y 1829:+ 1824:4 1820:x 1791:1 1788:= 1783:2 1779:y 1775:N 1767:2 1763:x 1751:. 1733:2 1729:b 1725:+ 1720:2 1716:a 1693:2 1689:b 1685:+ 1680:2 1676:a 1661:b 1657:a 1641:. 1636:p 1628:1 1620:1 1614:p 1610:a 1599:p 1595:a 1385:2 1379:m 1368:2 1364:a 1357:n 1335:1 1329:m 1318:1 1314:a 1307:n 1260:2 1256:w 1252:= 1249:) 1246:z 1243:, 1240:y 1237:, 1234:x 1231:( 1228:f 1206:2 1202:z 1198:= 1195:) 1192:y 1189:, 1186:x 1183:( 1180:f 1092:( 1030:, 1024:, 1019:5 1014:, 1009:3 898:( 802:2 769:2 705:a 701:/ 697:c 670:, 665:2 660:) 655:) 649:x 646:1 641:+ 638:x 634:( 628:2 625:1 619:( 614:= 611:1 608:+ 603:2 598:) 593:) 587:x 584:1 576:x 572:( 566:2 563:1 557:( 516:2 512:c 508:= 503:2 499:b 495:+ 490:2 486:a 465:) 462:c 459:, 456:b 453:, 450:a 447:( 431:( 261:e 254:t 247:v 43:. 36:. 20:)

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.