608:
divisions coming out even. That means that in the problem as stated, any multiple of 15,625 may be added to the pile, and it will satisfy the problem conditions. That also means that the number of coconuts in the original pile is smaller than 15,625, else subtracting 15,625 will yield a smaller solution. But the number in the original pile is not trivially small, like 5 or 10 (that is why this is a hard problem) – it may be in the hundreds or thousands. Unlike trial and error in the case of guessing a polynomial root, trial and error for a
Diophantine root will not result in any obvious convergence. There is no simple way of estimating what the solution will be.
450:. According to Gardner, Williams had modified an older problem to make it more confounding. In the older version there is a coconut for the monkey on the final division; in Williams's version the final division in the morning comes out even. But the available historical evidence does not indicate which versions Williams had access to. Gardner once told his son Jim that it was his favorite problem. He said that the Monkey and the Coconuts is "probably the most worked on and least often solved" Diophantine puzzle. Since that time the Williams version of the problem has become a staple of
6678:=0, so the initial pile contains -4 coconuts. This works because after tossing one positive coconut to the monkey, there are -5 coconuts in the pile. After division, there remain -4 coconuts. No matter how many such divisions are done, the remaining pile will contain -4 coconuts. This is a mathematical anomaly called a "fixed point". Only a few problems have such a point, but when there is one, it makes the problem much easier to solve. All solutions to the problem are multiples of 5 added to or subtracted from the fixed point.
617:
1054:
fifths such that each blue coconut is in a different fifth; then he takes the fifth with no blue coconut, gives one of his coconuts to the monkey, and puts the other four fifths (including all four blue coconuts) back together. Each sailor does the same. During the final division in the morning, the blue coconuts are left on the side, belonging to no one. Since the whole pile was evenly divided 5 times in the night, it must have contained 5 coconuts: 4 blue coconuts and 3121 ordinary coconuts.
1005:
66:
25:
357:, a historian of puzzles, traces a series of less plausibly related problems through the middle ages, with a few references as far back as the Babylonian empire circa 1700 BC. They involve the general theme of adding or subtracting fractions of a pile or specific numbers of discrete objects and asking how many there could have been in the beginning. The next reference to a similar problem is in
1068:: A man wills 17 horses to his three sons, specifying that the eldest son gets half, the next son one-third, and the youngest son, one-ninth of the animals. The sons are confounded, so they consult a wise horse trader. He says, "here, borrow my horse." The sons duly divide the horses, discovering that all the divisions come out even, with one horse left over, which they return to the trader.
168:
1040:)+1. If only one sailor woke up in the night, then 5/4(20)+1 = 26 works for the minimum number of coconuts in the original pile. But if two sailors woke up, 26 is not divisible by 4, so the morning pile must be some multiple of 20 that yields a pile divisible by 4 before the last sailor wakes up. It so happens that 3*20=60 works for two sailors: applying the recursion formula for
380:'s diaries in 1888: it involves a pile of nuts on a table serially divided by four brothers, each time with remainder of one given to a monkey, and the final division coming out even. The problem never appeared in any of Carroll's published works, though from other references it appears the problem was in circulation in 1888. An almost identical problem appeared in
1045:
obtuse, because in order to accommodate 4 sailors awakening, the morning pile must be some multiple of 60: if one is persistent, it may be discovered that 17*60=1020 does the trick and the minimum number in the original pile would be 2496. A last iteration on 2496 for 5 sailors awakening, i.e. 5/4(2496)+1 brings the original pile to 3121 coconuts.
1092:... represent the digits of the sailor's share S, both base 5. After the monkey's share, the least significant digit of N must now be 0; after the subtraction, the least significant digit of N' left by the first sailor must be 1, hence the following (the actual number of digits in N as well as S is unknown, but they are irrelevant just now):
1381:=1 yielding 21 is the smallest positive number that can be successively divided by 5 twice with remainder 1. If there are 5 divisions, then multiples of 5=3125 are required; the smallest such number is 3125 – 4 = 3121. After 5 divisions, there are 1020 coconuts left over, a number divisible by 5 as required by the problem. In fact, after
254:; the problem is to find the number of coconuts in the original pile (fractional coconuts not allowed). The problem is notorious for its confounding difficulty to unsophisticated puzzle solvers, though with the proper mathematical approach, the solution is trivial. The problem has become a staple in
2170:
The
Euclidean algorithm is quite tedious but a general methodology for solving rational equations ax+by=c requiring integral answers. From (2) above, it is evident that 1024 (2) and 15625 (5) are relatively prime and therefore their GCD is 1, but we need the reduction equations for back substitution
487:
is the study of equations with rational coefficients requiring integer solutions. In
Diophantine problems, there are fewer equations than unknowns. The "extra" information required to solve the equations is the condition that the solutions be integers. Any solution must satisfy all equations. Some
1044:
twice yields 96 as the smallest number of coconuts in the original pile. 96 is divisible by 4 once more, so for 3 sailors awakening, the pile could have been 121 coconuts. But 121 is not divisible by 4, so for 4 sailors awakening, one needs to make another leap. At this point, the analogy becomes
416:
By and by each of the five men woke up and did the same thing, one after the other: each one taking a fifth of the coconuts that were in the pile when he woke up, and having one left over for the monkey. In the morning they divided what coconuts were left, and they came out in five equal shares. Of
6422:
Other variants in which the number of men or the remainders vary between divisions, are generally outside the class of problems associated with the monkey and the coconuts, though these similarly reduce to linear
Diophantine equations in two variables. Their solutions yield to the same techniques
1020:
The search space can be reduced by a series of increasingly larger factors by observing the structure of the problem so that a bit of trial and error finds the solution. The search space is much smaller if one starts with the number of coconuts received by each man in the morning division, because
286:
Problems ask for either the initial or terminal quantity. Stated or implied is the smallest positive number that could be a solution. There are two unknowns in such problems, the initial number and the terminal number, but only one equation which is an algebraic reduction of an expression for the
278:
The monkey and the coconuts is the best known representative of a class of puzzle problems requiring integer solutions structured as recursive division or fractionating of some discretely divisible quantity, with or without remainders, and a final division into some number of equal parts, possibly
607:
Before entering upon a solution to the problem, a couple of things may be noted. If there were no remainders, given there are 6 divisions of 5, 5=15,625 coconuts must be in the pile; on the 6th and last division, each sailor receives 1024 coconuts. No smaller positive number will result in all 6
270:
There is a pile of coconuts, owned by five men. One man divides the pile into five equal piles, giving the one left over coconut to a passing monkey, and takes away his own share. The second man then repeats the procedure, dividing the remaining pile into five and taking away his share, as do the
1053:
Another device is to use extra objects to clarify the division process. Suppose that in the evening we add four blue coconuts to the pile. Then the first sailor to wake up will find the pile to be evenly divisible by five, instead of having one coconut left over. The sailor divides the pile into
1349:
A simple succinct solution can be obtained by directly utilizing the recursive structure of the problem: There were five divisions of the coconuts into fifths, each time with one left over (putting aside the final division in the morning). The pile remaining after each division must contain an
1526:
So if we began in modulo class –4 nuts then we will remain in modulo class –4. Since ultimately we have to divide the pile 5 times or 5^5, the original pile was 5^5 – 4 = 3121 coconuts. The remainder of 1020 coconuts conveniently divides evenly by 5 in the morning. This solution essentially
327:(“Compendium of the Essence of Mathematics”), circa 850CE, which dealt with serial division of fruit and flowers with specified remainders. That would make progenitor problems over 1000 years old before their resurgence in the modern era. Problems involving division which invoke the
1395:
The original pile of coconuts will be divided by 5 a total of 5 times with a remainder of 1, not considering the last division in the morning. Let N = number of coconuts in the original pile. Each division must leave the number of nuts in the same congruence class (mod 5). So,
1293:
are also 4, etc. There are 5 divisions; the first four must leave an odd number base 5 in the pile for the next division, but the last division must leave an even number base 5 so the morning division will come out even (in 5s). So there are four 4s in N following a LSD of 1:
2486:, rather than a computational value, careful algebraic reduction of the relation between the number of coconuts in the original pile and the number allotted to each sailor in the morning yields an analogous Diophantine relation whose coefficients are expressions in
5863:, the original pile has 1020 coconuts, and after four successive even divisions in the night with a coconut allocated to the monkey after each division, there are 80 coconuts left over in the morning, so the final division comes out even with no coconut left over.
287:
relation between them. Common to the class is the nature of the resulting equation, which is a linear
Diophantine equation in two unknowns. Most members of the class are determinate, but some are not (the monkey and the coconuts is one of the latter). Familiar
6146:, the original pile has 51 coconuts, and after three successive divisions in the night with a coconut allocated to the monkey after each division, there are 13 coconuts left over in the morning, so the final division has a coconut left over for the monkey.
3869:
3649:
3495:
3157:
848:
1204:
Since the next sailor is going to do the same thing on N', the least significant digit of N' becomes 0 after tossing one to the monkey, and the LSD of S' must be 4 for the same reason; the next digit of N' must also be 4. So now it looks like:
2160:
282:
Another example: "I have a whole number of pounds of cement, I know not how many, but after addition of a ninth and an eleventh, it was partitioned into 3 sacks, each with a whole number of pounds. How many pounds of cement did I have?"
271:
third, fourth, and fifth, each of them finding one coconut left over when dividing the pile by five, and giving it to a monkey. Finally, the group divide the remaining coconuts into five equal piles: this time no coconuts are left over.
5750:
The alternate form also had two endings, when the morning division comes out even, and when there is one nut left over for the monkey. When the morning division comes out even, the general solution reduces via a similar derivation to:
1766:
434:, famously fired off a telegram to Williams saying: "FOR THE LOVE OF MIKE, HOW MANY COCONUTS? HELL POPPING AROUND HERE". Williams continued to get letters asking for a solution or proposing new ones for the next twenty years.
1873:
to be an integer, the RHS of the equation must be an integral multiple of 1024; that property will be unaltered by factoring out as many multiples of 1024 as possible from the RHS. Reducing both sides by multiples of 1024,
410:
During the night, one man woke up, and decided to take his share early. So he divided the coconuts in five piles. He had one coconut left over, and he gave that to the monkey. Then he hid his pile and put the rest back
1588:
is the number received by each sailor on the final division in the morning. This is only trivially different than the equation above for the predecessor problem, and solvability is guaranteed by the same reasoning.
744:
2957:
4027:
1958:
339:
first studied problems requiring integer solutions in the 3rd century CE. The
Euclidean algorithm for greatest common divisor which underlies the solution of such problems was discovered by the Greek geometer
6245:
5095:=–1, is –12504, so trial and error around zero will not solve the Williams version of the problem, unlike the original version whose equation, fortuitously, had a small magnitude negative solution).
4819:
2452:
5944:
4298:
6967:
4201:
2613:
6864:
5745:
4590:
5528:
279:
with a remainder. The problem is so well known that the entire class is often referred to broadly as "monkey and coconut type problems", though most are not closely related to the problem.
6149:
Other post-Williams variants which specify different remainders including positive ones (i.e. the monkey adds coconuts to the pile), have been treated in the literature. The solution is:
4411:
2304:
2655:
2356:
1848:
627:'s 1958 Mathematical Games column begins its analysis by solving the original problem (with one coconut also remaining in the morning) because it is easier than Williams's version. Let
2070:
5466:
3727:
5610:
5410:
1449:
3506:
7004:=3 satisfies the equation, 11 is the smallest positive number that gives each sailor a non-zero positive number of coconuts on each division, an implicit condition of the problem.
5806:
2008:
5360:
3288:
2968:
2688:
1076:
A simple solution appears when the divisions and subtractions are performed in base 5. Consider the subtraction, when the first sailor takes his share (and the monkey's). Let n
893:
749:
6746:
1631:
1574:
631:
be the number of coconuts received by each sailor after the final division into 5 equal shares in the morning. Then the number of coconuts left before the morning division is
4707:
5316:
3280:
3221:
994:
6535:
Antonick (2013): "I then asked Jim if his father had a favorite puzzle, and he answered almost immediately: 'The monkeys and the coconuts. He was quite fond of that one.'"
1489:
5278:
5240:
4342:
1520:
948:
6337:
6118:
5983:
4911:
6281:
6036:
4855:
2748:
2093:
5061:
4524:
4443:
3899:
3679:
2715:
2534:
658:
388:(1890). The problem was mentioned in works of period mathematicians, with solutions, mostly wrong, indicating that the problem was new and unfamiliar at the time.
6144:
6082:
5861:
5835:
5665:
5639:
4492:
6989:
6886:
6789:
6766:
6417:
6397:
6377:
6357:
6301:
6056:
6003:
5202:
5181:
5156:
5136:
5116:
5081:
5034:
5011:
4991:
4971:
4951:
4931:
4875:
4727:
4466:
4221:
4133:
4110:
4090:
4070:
4050:
3919:
3719:
3699:
2767:
2504:
2484:
2250:
1871:
1790:
1679:
1032:, which must also be divisible by 4, since the last sailor in the night combined 4 piles for the morning division. So the morning pile, call the number
426:
Williams had not included an answer in the story. The magazine was inundated by more than 2,000 letters pleading for an answer to the problem. The
1337:
Since that number must be an integer and 1024 is relatively prime to 3125, N+4 must be a multiple of 3125. The smallest such multiple is 3125
585:
is a parameter than can be any integer. The problem is not intended to be solved by trial-and-error; there are deterministic methods for solving (
186:
663:
2775:
5091: 5 – 4 = 3121 for the number of coconuts in the original pile. (It may be noted that the next sequential solution of the equation for
83:
38:
3927:
1880:
130:
2509:
The first step is to obtain an algebraic expansion of the recurrence relation corresponding to each sailor's transformation of the pile,
6621:
S. Singh and D. Bhattacharya, “On
Dividing Coconuts: A Linear Diophantine Problem,” The College Mathematics Journal, May 1997, pp. 203–4
5670:
In some earlier alternate forms of the problem, the divisions came out even, and nuts (or items) were allocated from the remaining pile
102:
6612:
Kirchner, Roger B. "The
Generalized Coconut Problem," The American Mathematical Monthly 67, no. 6 (1960): 516-19. doi:10.2307/2309167.
1309:
A straightforward numeric analysis goes like this: If N is the initial number, each of 5 sailors transitions the original pile thus:
6579:
Pappas, T. "The Monkey and the
Coconuts." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 226-227 and 234, 1989.
6155:
1369: 5=25 rather than 5 must be used, because 25 can be divided by 5 twice. So the number of coconuts that could be in the pile is
109:
2366:
Alternately, one may use a continued fraction, whose construction is based on the
Euclidean algorithm. The continued fraction for
898:
Gardner points out that this equation is "much too difficult to solve by trial and error," but presents a solution he credits to
6653:
417:
course each one must have known there were coconuts missing; but each one was guilty as the others, so they didn't say anything.
6603:
Underwood, R. S., and Robert E. Moritz. "3242." The American Mathematical Monthly 35, no. 1 (1928): 47-48. doi:10.2307/2298601.
5540:
Other variants, including the putative predecessor problem, have related general solutions for an arbitrary number of sailors.
1057:
The device of using additional objects to aid in conceptualizing a division appeared as far back as 1912 in a solution due to
4735:
116:
2404:
1341: 1, so N = 3125 – 4 = 3121; the number left in the morning comes to 1020, which is evenly divisible by 5 as required.
604:
Numerous solutions starting as early as 1928 have been published both for the original problem and Williams modification.
6553:
5872:
4229:
465:
1354: 1+1=6 is a solution. In fact any multiple of five plus one is a solution, so a possible general formula is 5
98:
6891:
4138:
2542:
7122:
6794:
6511:
5680:
4532:
488:
Diophantine equations have no solution, some have one or a finite number, and others have infinitely many solutions.
222:
204:
149:
52:
5477:
4092:
is a parameter that can be any integer. The nature of the equation and the method of its solution do not depend on
44:
4350:
2258:
3864:{\displaystyle m\cdot F=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-(m-1)\cdot {\frac {m^{m}-(m-1)^{m}}{m^{m}}}}
1028:
is the number of coconuts each sailor receives in the final division in the morning, the pile in the morning is 5
2621:
2310:
1806:
7137:
7132:
7053:
6432:
2019:
87:
5421:
3644:{\displaystyle n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-(m-1)\cdot {\frac {m^{m}-(m-1)^{m}}{m^{m}}}}
1362:– 4, since a multiple of 5 plus 1 is also a multiple of 5 minus 4. So 11, 16, etc also work for one division.
906:): The equation also has solutions in negative integers. Trying out a few small negative numbers it turns out
5667:
this yields 15,621 as the smallest positive number of coconuts for the pre-William's version of the problem.
5549:
300:
5371:
1402:
1350:
integral number of coconuts. If there were only one such division, then it is readily apparent that 5
3490:{\displaystyle n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-\left({\frac {m-1}{m}}\right)\cdot \left}
3152:{\displaystyle n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-\left({\frac {m-1}{m}}\right)\cdot \left}
1065:
843:{\displaystyle {\tfrac {5}{4}}({\tfrac {25}{4}}F+{\tfrac {9}{4}})+1={\tfrac {125}{16}}F+{\tfrac {241}{16}}}
471:
Numerous variants which vary the number of sailors, monkeys, or coconuts have appeared in the literature.
405:
Five men and a monkey were shipwrecked on an island. They spent the first day gathering coconuts for food.
6688:
5757:
1969:
370:
123:
5327:
2660:
860:
6703:
2221:
1 = –81·1024 + 313·(15625 – 15·1024) = 313·15625 – 4776·1024 (substitute for 265 from (a) and combine)
1598:
1541:
395:
modified an older problem and included it in a story, "Coconuts", in the October 9, 1926, issue of the
6700:
Gardner gives an equivalent but rather cryptic formulation by inexplicably choosing the non-canonical
6565:
4667:
335:
asked: Find a number which leaves the remainders 2, 3 and 2 when divided by 3, 5 and 7, respectively.
5289:
3226:
3169:
441:
328:
182:
1688:
961:
6438:
1456:
451:
336:
255:
5251:
7071:
5213:
4306:
512:
460:
76:
2155:{\displaystyle 1024N=15625\cdot 204+8404\Rightarrow N={\frac {3195904}{1024}}\Rightarrow N=3121}
1496:
524:
3437:
2218:
1 = –81·(1024 – 3·265) + 70·265 = –81·1024 + 313·265 (substitute for 229 from (b) and combine)
909:
6306:
6087:
5952:
4880:
1285:
But the same reasoning again applies to N' as applied to N, so the next digit of N' is 4, so s
491:
The monkey and the coconuts reduces to a two-variable linear Diophantine equation of the form
7083:
6630:
G. Salvatore and T. Shima, "Of coconuts and integrity," Crux Mathematicorum, 4 (1978) 182–185
6592:
6253:
6008:
4827:
2720:
431:
397:
7060:
6501:
1012:
Trial and error fails to solve Williams's version, so a more systematic approach is needed.
7021:
5039:
4497:
4416:
3877:
3657:
2693:
2512:
2215:
1 = –11·229 + 70·(265 – 1·229) = –81·229 + 70·265 (substitute for 36 from (c) and combine)
484:
346:
243:
634:
8:
7078:
7066:
Kirchner, R. B. "The Generalized Coconut Problem." Amer. Math. Monthly 67, 516-519, 1960.
7034:
6123:
6061:
5840:
5814:
5644:
5618:
4471:
1638:
446:
299:
The origin of the class of such problems has been attributed to the Indian mathematician
239:
6974:
6871:
6774:
6751:
6444:
6402:
6382:
6362:
6342:
6286:
6041:
5988:
5187:
5166:
5141:
5121:
5101:
5066:
5019:
4996:
4976:
4956:
4936:
4916:
4860:
4712:
4451:
4206:
4118:
4095:
4075:
4055:
4035:
3904:
3704:
3684:
2752:
2489:
2469:
2252:(see (3) in the previous subsection) that will make both N and F positive is 2569, so:
2235:
2206:
1 = 10 – 3(13–1·10) = 4·10 – 3·13 (reorder (g), substitute for 3 from (f) and combine)
1856:
1775:
899:
2212:
1 = 4·36 – 11·(229 – 6·36) = –11·229 + 70*36 (substitute for 13 from (d) and combine)
7142:
7049:
6507:
6462:
392:
7127:
6649:
2458:
This is the smallest positive number that satisfies the conditions of the problem.
1761:{\displaystyle {\begin{cases}N=N_{0}+15625\cdot t\\F=F_{0}+1024\cdot t\end{cases}}}
1058:
1036:, is a multiple of 20. The pile before the last sailor woke up must have been 5/4(
354:
1641:; in fact, it has infinitely many periodic solutions positive and negative. If (x
1084:,... represent the digits of N, the number of coconuts in the original pile, and s
616:
6768:
is even, then refactoring the expression in a way that obscures the periodicity:
2209:
1 = 4·(36 – 2·13) – 3·13 = 4·36 – 11·13 (substitute for 10 from (e) and combine)
2075:
The RHS must still be a multiple of 1024; since 53 is relatively prime to 1024, 5
455:
381:
6497:
5866:
When the morning division results in a nut left over, the general solution is:
624:
437:
376:
The first description of the problem in close to its modern wording appears in
358:
7103:
7097:
535:. If it does, the equation has infinitely many periodic solutions of the form
401:. Here is how the problem was stated by Williams (condensed and paraphrased):
7116:
7041:
7016:
377:
1673: 8404 is a solution of (2), which means any solution must have the form
3163:
391:
The problem became notorious when American novelist and short story writer
2690:
the number left in the morning. Expanding the recurrence by substituting
6570:, by Clifton Fadiman, Mathematical Association of America, Springer, 1997
1637:
This Diophantine equation has a solution which follows directly from the
739:{\displaystyle {\tfrac {5}{4}}(5F+1)+1={\tfrac {25}{4}}F+{\tfrac {9}{4}}}
5543:
When the morning division also has a remainder of one, the solution is:
2952:{\displaystyle n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-\left}
2376:(0.065536 exactly) is ; its convergent terminated after the repetend is
1317:
Repeating this transition 5 times gives the number left in the morning:
903:
1004:
454:. The original story containing the problem was reprinted in full in
2079:+4 must be a multiple of 1024. The smallest such multiple is 1
4022:{\displaystyle N=m^{m}\cdot {\frac {m-1+m\cdot F}{(m-1)^{m}}}-(m-1)}
1953:{\displaystyle 0=(15625F-15\cdot 1024F)+(8404-8\cdot 1024)\mod 1024}
1385:
divisions, it can be proven that the remaining pile is divisible by
65:
1389:, a property made convenient use of by the creator of the problem.
468:
recommends for acquisition by undergraduate mathematics libraries.
366:
1021:
that number is much smaller than the number in the original pile.
5083:
may be taken to be zero to obtain the lowest positive answer, so
1530:
332:
331:
appeared in Chinese literature as early as the first century CE.
288:
251:
6379:
is the remainder after each division (or number of monkeys) and
4032:
The equation is a linear Diophantine equation in two variables,
341:
247:
6240:{\displaystyle N=r_{0}\cdot m^{m}-c\cdot (m-1)+k\cdot m^{m+1}}
246:
that originated in a short story involving five sailors and a
1792:
is an arbitrary parameter that can have any integral value.
1754:
7046:
The Joy of Mathematics: Discovering Mathematics All Around
6463:
Chronology of Recreational Mathematics by David Singmaster
4526:
by Bézout's identity. This equation can be restated as:
3681:
is the number left in the morning which is a multiple of
2203:
10 = 3·3 + 1 (g) (remainder 1 is GCD of 15625 and 1024)
1535:
The equivalent Diophantine equation for this version is:
1150:
The digit subtracted from 0 base 5 to yield 1 is 4, so s
854:
of the original pile satisfies the Diophantine equation
5535:
4814:{\displaystyle N=r_{0}\cdot m^{m}-(m-1)+k\cdot m^{m+1}}
2182:
First, obtain successive remainders until GCD remains:
1313:
N => 4(N–1)/5 or equivalently, N => 4(N+4)/5 – 4.
6441:, possibly the most famous Diophantine equation of all
6419:
is negative if the monkeys add coconuts to the pile).
5674:
division. In these forms, the recursion relation is:
3721:, the number allotted to each sailor in the morning):
2466:
When the number of sailors is a parameter, let it be
2447:{\displaystyle N=-4776\cdot 8404+15625\cdot 2569=3121}
1853:
Another way of thinking about it is that in order for
829:
811:
787:
769:
754:
746:; the number present when the fourth sailor awoke was
725:
707:
668:
6977:
6894:
6874:
6797:
6777:
6754:
6706:
6405:
6385:
6365:
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6309:
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6158:
6126:
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6044:
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5991:
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2624:
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2407:
2313:
2261:
2238:
2096:
2022:
1972:
1883:
1859:
1809:
1800:
One can take both sides of (1) above modulo 1024, so
1778:
1682:
1601:
1544:
1527:
reverses how the problem was (probably) constructed.
1499:
1459:
1405:
964:
912:
863:
752:
666:
660:; the number present when the fifth sailor awoke was
637:
373:
and applied it to solution of Diophantine equations.
6435:, a substantially more difficult Diophantine problem
5013:
satisfies the constraints of the problem statement.
4494:
are relatively prime, there exist integer solutions
2398:
for which both N and F are non-negative is 2569, so
5939:{\displaystyle N=r_{0}\cdot m^{m}-m+k\cdot m^{m+1}}
4293:{\displaystyle r={\frac {m-1+m\cdot F}{(m-1)^{m}}}}
291:methods are unavailing for solving such equations.
177:
may be too technical for most readers to understand
90:. Unsourced material may be challenged and removed.
6983:
6961:
6880:
6858:
6783:
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6295:
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5997:
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4084:
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3489:
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3215:
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2002:
1952:
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1784:
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1625:
1568:
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988:
942:
887:
842:
738:
652:
422:How many coconuts were there in the original pile?
274:How many coconuts were there in the original pile?
6962:{\displaystyle N=(m-1+m\cdot k)\cdot m^{m}-(m-1)}
4196:{\displaystyle {\frac {m-1+m\cdot F}{(m-1)^{m}}}}
2608:{\displaystyle n_{i}={\frac {m-1}{m}}(n_{i-1}-1)}
1451:(mod 5) (the –1 is the nut tossed to the monkey)
1158:is just shifting the number right one position, n
7114:
7035:Problem of the Week: The Monkey and the Coconuts
6648:
4115:Number theoretic considerations now apply. For
1365:If two divisions are done, a multiple of 5
6859:{\displaystyle N=(1+m\cdot k)\cdot m^{m}-(m-1)}
5740:{\displaystyle n_{i}={\frac {m-1}{m}}n_{i-1}-1}
4585:{\displaystyle (m-1)^{m}\cdot r\equiv -1\mod m}
4303:The equation must be transformed into the form
1392:A formal way of stating the above argument is:
6496:
5523:{\displaystyle 117,440,505+k\cdot 134,217,728}
4664:Bézout's identity gives the periodic solution
1531:The Diophantine equation and forms of solution
1154:=4. But since S is (N-1)/5, and dividing by 5
1008:Williams's version, with no coconuts left over
6492:
6490:
6488:
6486:
6484:
6482:
4729:in the Diophantine equation and rearranging:
7106:, a copy of the story as it appeared in the
7022:Martin Gardner’s The Monkey and the Coconuts
4406:{\displaystyle (m-1)^{m}\cdot r-m\cdot s=-1}
2299:{\displaystyle N=N_{0}+15625\cdot 2569=3121}
620:Original version, with one coconut left over
6506:. W.W. Norton & Company. pp. 3–9.
250:on a desert island who divide up a pile of
53:Learn how and when to remove these messages
6642:
6479:
6472:
6470:
5098:Here is a table of the positive solutions
2650:{\displaystyle n_{i\rightarrow 0}\equiv N}
2351:{\displaystyle F=F_{0}+1024\cdot 2569=204}
2232:) = (-4776·8404, -313*8404); the smallest
2087:+4=1024 and F=204. Substituting into (1)
1843:{\displaystyle 1024N=15625F+8404\mod 1024}
1795:
1333: => 1024(N+4)/3125 – 4
527:, the equation is solvable if and only if
365:, 1725. In the realm of pure mathematics,
7048:Wide World Publishing, January 23, 1993,
5016:In the William's version of the problem,
4578:
4577:
2461:
2065:{\displaystyle 0=53\cdot (5F+4)\mod 1024}
2058:
2057:
1996:
1995:
1946:
1945:
1836:
1835:
1649:) is a solution of 1024x–15625y=1, then N
223:Learn how and when to remove this message
205:Learn how and when to remove this message
189:, without removing the technical details.
150:Learn how and when to remove this message
6554:KIRKUS REVIEW of The Mathematical Magpie
5461:{\displaystyle 823,537+k\cdot 5,764,801}
4135:to be an integer, it is sufficient that
1584:is the original number of coconuts, and
1330: => 256(N+4)/625 – 4
1003:
615:
6991:is a parameter than can be any integer.
6467:
5605:{\displaystyle N=-(m-1)+k\cdot m^{m+1}}
4635:is a polynomial with monomial basis in
1327: => 64(N+4)/125 – 4
1304:
1166:=4. So now the subtraction looks like:
958:to get the smallest positive solution:
611:
440:featured the problem in his April 1958
7115:
5405:{\displaystyle 233,275+k\cdot 279,936}
4344:whose solutions are formulaic. Hence:
2657:, the number originally gathered, and
2165:
1444:{\displaystyle N\equiv 4/5\cdot (N-1)}
1324: => 16(N+4)/25 – 4
363:Récréations mathématiques et physiques
261:
2536:being the number left by the sailor:
2361:
187:make it understandable to non-experts
5536:Other variants and general solutions
1522:(mod 5) (–4 is the congruence class)
1344:
161:
88:adding citations to reliable sources
59:
18:
5801:{\displaystyle N=-m+k\cdot m^{m+1}}
3166:polynomial in brackets of the form
2003:{\displaystyle 0=265F+212\mod 1024}
1071:
999:
850:; and so on. We find that the size
466:Mathematical Association of America
13:
7098:Monkeys and Coconuts – Numberphile
5355:{\displaystyle 3121+k\cdot 15,625}
2683:{\displaystyle n_{i\rightarrow m}}
2179:in terms of these two quantities:
888:{\displaystyle 1024N=15625F+11529}
14:
7154:
7091:
6741:{\displaystyle r_{0}=-1+1\cdot m}
6423:and present no new difficulties.
1626:{\displaystyle 1024N-15625F=8404}
1569:{\displaystyle 1024N=15625F+8404}
266:The problem can be expressed as:
34:This article has multiple issues.
6691:for an exposition of the method.
6503:The Colossal Book of Mathematics
4702:{\displaystyle r=r_{0}+k\cdot m}
1255:as well. So now it looks like:
1064:A related device appears in the
1048:
1015:
166:
64:
23:
6994:
6694:
6681:
6668:
6633:
6624:
6615:
6606:
6597:
6582:
5311:{\displaystyle 765+k\cdot 1024}
4573:
3275:{\displaystyle (1-x^{m})/(1-x)}
3216:{\displaystyle x^{m-1}+...+x+1}
2053:
1991:
1941:
1831:
1247:(which is now 4) leaves 3, so s
950:is a solution. We add 15625 to
75:needs additional citations for
42:or discuss these issues on the
6956:
6944:
6925:
6901:
6853:
6841:
6822:
6804:
6658:School Science and Mathematics
6591:can be found if necessary via
6573:
6559:
6547:
6538:
6529:
6520:
6456:
6209:
6197:
5574:
5562:
5158:is any non-negative integer):
4783:
4771:
4549:
4536:
4513:
4501:
4367:
4354:
4278:
4265:
4181:
4168:
4016:
4004:
3989:
3976:
3839:
3826:
3804:
3792:
3619:
3606:
3584:
3572:
3474:
3453:
3421:
3399:
3269:
3257:
3249:
3230:
3090:
3068:
2914:
2892:
2868:
2846:
2672:
2633:
2602:
2577:
2140:
2121:
2050:
2035:
1938:
1920:
1914:
1890:
1438:
1426:
989:{\displaystyle N=15621,F=1023}
798:
765:
694:
679:
1:
6450:
1484:{\displaystyle 5N\equiv 4N-4}
99:"The monkey and the coconuts"
5273:{\displaystyle 25+k\cdot 81}
4953:is any integer. For a given
4203:be an integer, so let it be
1066:17-animal inheritance puzzle
474:
7:
6654:"Problem Department (#288)"
6433:Archimedes's cattle problem
6426:
6058:is even. For example, when
5235:{\displaystyle 11+k\cdot 8}
4337:{\displaystyle ax+by=\pm 1}
2962:Factoring the latter term,
599:) in this case (see text).
236:The monkey and the coconuts
10:
7159:
7061:Monkey and Coconut Problem
7010:
2394:=313. The least value of
2185:15625 = 15·1024 + 265 (a)
1515:{\displaystyle N\equiv -4}
1251:must be 4, and therefore n
371:continued fraction theorem
294:
7069:Fadiman, Clifton (1962).
4993:will be chosen such that
943:{\displaystyle N=-4,F=-1}
442:Mathematical Games column
329:Chinese remainder theorem
303:in chapter VI, § 131
7123:Recreational mathematics
7032:Pleacher, David (2005).
6332:{\displaystyle r_{0}=-1}
6113:{\displaystyle r_{0}=-1}
5978:{\displaystyle r_{0}=-1}
4973:, the smallest positive
4906:{\displaystyle r_{0}=-1}
452:recreational mathematics
337:Diophantus of Alexandria
256:recreational mathematics
7072:The Mathematical Magpie
6674:A special case is when
6567:The Mathematical Magpie
6276:{\displaystyle r_{0}=1}
6031:{\displaystyle r_{0}=1}
4850:{\displaystyle r_{0}=1}
4661:is even is a solution.
2743:{\displaystyle n_{i-1}}
2188:1024 = 3·265 + 229 (b)
1796:A reductionist approach
513:greatest common divisor
461:The Mathematical Magpie
7075:, Simon & Schuster
6985:
6963:
6882:
6860:
6785:
6762:
6742:
6413:
6393:
6373:
6353:
6333:
6297:
6277:
6241:
6140:
6114:
6078:
6052:
6032:
5999:
5979:
5940:
5857:
5831:
5802:
5741:
5661:
5635:
5606:
5524:
5462:
5406:
5356:
5312:
5274:
5236:
5198:
5177:
5152:
5132:
5112:
5077:
5057:
5030:
5007:
4987:
4967:
4947:
4927:
4907:
4871:
4851:
4815:
4723:
4709:, so substituting for
4703:
4586:
4520:
4488:
4462:
4439:
4407:
4338:
4294:
4217:
4197:
4129:
4106:
4086:
4066:
4046:
4023:
3915:
3895:
3865:
3715:
3695:
3675:
3645:
3491:
3276:
3217:
3153:
2953:
2763:
2744:
2711:
2684:
2651:
2609:
2530:
2500:
2480:
2462:A generalized solution
2448:
2352:
2300:
2246:
2156:
2066:
2004:
1954:
1867:
1844:
1786:
1762:
1627:
1570:
1516:
1485:
1445:
1009:
990:
944:
889:
844:
740:
654:
621:
601:
369:in 1770 expounded his
7138:Diophantine equations
7133:Mathematical problems
7108:Saturday Evening Post
6986:
6964:
6883:
6861:
6786:
6763:
6743:
6439:Fermat's Last Theorem
6414:
6394:
6374:
6354:
6334:
6298:
6278:
6242:
6141:
6115:
6079:
6053:
6033:
6000:
5980:
5941:
5858:
5832:
5803:
5742:
5662:
5636:
5607:
5525:
5463:
5407:
5357:
5313:
5275:
5237:
5199:
5178:
5153:
5133:
5113:
5078:
5058:
5056:{\displaystyle r_{0}}
5031:
5008:
4988:
4968:
4948:
4928:
4908:
4872:
4852:
4816:
4724:
4704:
4587:
4521:
4519:{\displaystyle (r,s)}
4489:
4463:
4440:
4438:{\displaystyle s=1+F}
4408:
4339:
4295:
4218:
4198:
4130:
4107:
4087:
4067:
4047:
4024:
3916:
3896:
3894:{\displaystyle n_{0}}
3866:
3716:
3696:
3676:
3674:{\displaystyle n_{m}}
3646:
3500:which simplifies to:
3492:
3277:
3218:
3154:
2954:
2764:
2745:
2712:
2710:{\displaystyle n_{i}}
2685:
2652:
2610:
2531:
2529:{\displaystyle n_{i}}
2501:
2481:
2449:
2353:
2301:
2247:
2191:265 = 1·229 + 36 (c)
2157:
2067:
2005:
1955:
1868:
1845:
1787:
1763:
1628:
1571:
1517:
1486:
1446:
1007:
991:
945:
890:
845:
741:
655:
619:
502:(a/d)x + (b/d)y = c/d
482:
480:A Diophantine problem
398:Saturday Evening Post
344:and published in his
7079:Bogomolny, Alexander
6975:
6892:
6872:
6795:
6775:
6752:
6704:
6403:
6383:
6363:
6343:
6307:
6287:
6254:
6156:
6124:
6088:
6062:
6042:
6009:
5989:
5953:
5873:
5841:
5815:
5758:
5681:
5645:
5619:
5550:
5478:
5422:
5372:
5328:
5290:
5252:
5214:
5188:
5167:
5142:
5122:
5102:
5067:
5040:
5020:
4997:
4977:
4957:
4937:
4917:
4881:
4861:
4828:
4736:
4713:
4668:
4599:–1) is a polynomial
4533:
4498:
4472:
4452:
4417:
4351:
4307:
4230:
4207:
4139:
4119:
4096:
4076:
4056:
4036:
3928:
3905:
3878:
3728:
3705:
3685:
3658:
3507:
3289:
3227:
3170:
2969:
2776:
2753:
2721:
2694:
2661:
2622:
2543:
2513:
2490:
2470:
2405:
2311:
2259:
2236:
2194:229 = 6·36 + 13 (d)
2094:
2020:
1970:
1881:
1857:
1807:
1776:
1680:
1599:
1542:
1497:
1457:
1403:
1321:N => 4(N+4)/5 – 4
1305:A numerical approach
962:
910:
861:
750:
664:
653:{\displaystyle 5F+1}
635:
612:The original version
581:) is a solution and
498:, or more generally,
485:Diophantine analysis
325:Ganita-sara-sangraha
244:Diophantine analysis
84:improve this article
7058:Wolfram Mathworld:
6139:{\displaystyle k=1}
6077:{\displaystyle m=3}
5856:{\displaystyle k=1}
5830:{\displaystyle m=4}
5660:{\displaystyle k=1}
5634:{\displaystyle m=5}
4487:{\displaystyle m-1}
2197:36 = 2·13 + 10 (e)
2166:Euclidean algorithm
1639:Euclidean algorithm
1143:) 1 (N'
447:Scientific American
262:General description
240:mathematical puzzle
16:Mathematical puzzle
7029::, October 7, 2013
7025:in Numberplay The
6981:
6959:
6878:
6856:
6781:
6758:
6738:
6593:Euclid's algorithm
6445:Cannonball problem
6409:
6389:
6369:
6349:
6329:
6293:
6273:
6237:
6136:
6110:
6074:
6048:
6028:
5995:
5975:
5936:
5853:
5827:
5811:For example, when
5798:
5737:
5657:
5631:
5602:
5520:
5458:
5402:
5352:
5308:
5270:
5232:
5194:
5173:
5148:
5128:
5118:for the first few
5108:
5073:
5053:
5026:
5003:
4983:
4963:
4943:
4923:
4903:
4867:
4847:
4811:
4719:
4699:
4582:
4516:
4484:
4458:
4435:
4403:
4334:
4290:
4213:
4193:
4125:
4102:
4082:
4062:
4042:
4019:
3911:
3891:
3861:
3711:
3691:
3671:
3641:
3487:
3272:
3213:
3149:
2949:
2759:
2740:
2707:
2680:
2647:
2605:
2526:
2496:
2476:
2444:
2362:Continued fraction
2348:
2296:
2242:
2200:13 = 1·10 + 3 (f)
2152:
2062:
2000:
1950:
1863:
1840:
1782:
1758:
1753:
1623:
1566:
1512:
1481:
1441:
1243:Borrowing 1 from n
1010:
986:
940:
900:J. H. C. Whitehead
885:
840:
838:
820:
796:
778:
763:
736:
734:
716:
677:
650:
622:
464:, a book that the
458:'s 1962 anthology
386:Elementary Algebra
7084:Negative Coconuts
6984:{\displaystyle k}
6881:{\displaystyle m}
6784:{\displaystyle m}
6761:{\displaystyle m}
6544:Wolfram Mathworld
6412:{\displaystyle c}
6392:{\displaystyle k}
6372:{\displaystyle c}
6352:{\displaystyle m}
6296:{\displaystyle m}
6051:{\displaystyle m}
5998:{\displaystyle m}
5713:
5533:
5532:
5197:{\displaystyle N}
5176:{\displaystyle m}
5151:{\displaystyle k}
5131:{\displaystyle m}
5111:{\displaystyle N}
5076:{\displaystyle k}
5036:is 5 sailors, so
5029:{\displaystyle m}
5006:{\displaystyle N}
4986:{\displaystyle k}
4966:{\displaystyle m}
4946:{\displaystyle k}
4926:{\displaystyle m}
4870:{\displaystyle m}
4722:{\displaystyle r}
4461:{\displaystyle m}
4288:
4216:{\displaystyle r}
4191:
4128:{\displaystyle N}
4105:{\displaystyle m}
4085:{\displaystyle m}
4065:{\displaystyle F}
4045:{\displaystyle N}
3999:
3914:{\displaystyle N}
3859:
3764:
3714:{\displaystyle F}
3694:{\displaystyle m}
3639:
3544:
3472:
3418:
3374:
3326:
3136:
3087:
3054:
3006:
2942:
2911:
2865:
2813:
2762:{\displaystyle m}
2575:
2499:{\displaystyle m}
2479:{\displaystyle m}
2245:{\displaystyle t}
2138:
1866:{\displaystyle n}
1785:{\displaystyle t}
1345:Modulo congruence
837:
819:
795:
777:
762:
733:
715:
676:
525:Bézout's identity
393:Ben Ames Williams
233:
232:
225:
215:
214:
207:
160:
159:
152:
134:
57:
7150:
7005:
6998:
6992:
6990:
6988:
6987:
6982:
6968:
6966:
6965:
6960:
6940:
6939:
6887:
6885:
6884:
6879:
6865:
6863:
6862:
6857:
6837:
6836:
6791:odd,
6790:
6788:
6787:
6782:
6767:
6765:
6764:
6759:
6747:
6745:
6744:
6739:
6716:
6715:
6698:
6692:
6685:
6679:
6672:
6666:
6665:
6650:Norman H. Anning
6646:
6640:
6639:Bogomolny (1996)
6637:
6631:
6628:
6622:
6619:
6613:
6610:
6604:
6601:
6595:
6586:
6580:
6577:
6571:
6563:
6557:
6551:
6545:
6542:
6536:
6533:
6527:
6524:
6518:
6517:
6494:
6477:
6474:
6465:
6460:
6418:
6416:
6415:
6410:
6399:is any integer (
6398:
6396:
6395:
6390:
6378:
6376:
6375:
6370:
6358:
6356:
6355:
6350:
6338:
6336:
6335:
6330:
6319:
6318:
6302:
6300:
6299:
6294:
6282:
6280:
6279:
6274:
6266:
6265:
6246:
6244:
6243:
6238:
6236:
6235:
6187:
6186:
6174:
6173:
6145:
6143:
6142:
6137:
6119:
6117:
6116:
6111:
6100:
6099:
6083:
6081:
6080:
6075:
6057:
6055:
6054:
6049:
6037:
6035:
6034:
6029:
6021:
6020:
6004:
6002:
6001:
5996:
5984:
5982:
5981:
5976:
5965:
5964:
5945:
5943:
5942:
5937:
5935:
5934:
5904:
5903:
5891:
5890:
5862:
5860:
5859:
5854:
5836:
5834:
5833:
5828:
5807:
5805:
5804:
5799:
5797:
5796:
5746:
5744:
5743:
5738:
5730:
5729:
5714:
5709:
5698:
5693:
5692:
5666:
5664:
5663:
5658:
5640:
5638:
5637:
5632:
5611:
5609:
5608:
5603:
5601:
5600:
5529:
5527:
5526:
5521:
5467:
5465:
5464:
5459:
5411:
5409:
5408:
5403:
5361:
5359:
5358:
5353:
5317:
5315:
5314:
5309:
5279:
5277:
5276:
5271:
5241:
5239:
5238:
5233:
5203:
5201:
5200:
5195:
5182:
5180:
5179:
5174:
5161:
5160:
5157:
5155:
5154:
5149:
5137:
5135:
5134:
5129:
5117:
5115:
5114:
5109:
5082:
5080:
5079:
5074:
5062:
5060:
5059:
5054:
5052:
5051:
5035:
5033:
5032:
5027:
5012:
5010:
5009:
5004:
4992:
4990:
4989:
4984:
4972:
4970:
4969:
4964:
4952:
4950:
4949:
4944:
4932:
4930:
4929:
4924:
4912:
4910:
4909:
4904:
4893:
4892:
4876:
4874:
4873:
4868:
4856:
4854:
4853:
4848:
4840:
4839:
4820:
4818:
4817:
4812:
4810:
4809:
4767:
4766:
4754:
4753:
4728:
4726:
4725:
4720:
4708:
4706:
4705:
4700:
4686:
4685:
4591:
4589:
4588:
4583:
4557:
4556:
4525:
4523:
4522:
4517:
4493:
4491:
4490:
4485:
4467:
4465:
4464:
4459:
4444:
4442:
4441:
4436:
4412:
4410:
4409:
4404:
4375:
4374:
4343:
4341:
4340:
4335:
4299:
4297:
4296:
4291:
4289:
4287:
4286:
4285:
4263:
4240:
4222:
4220:
4219:
4214:
4202:
4200:
4199:
4194:
4192:
4190:
4189:
4188:
4166:
4143:
4134:
4132:
4131:
4126:
4111:
4109:
4108:
4103:
4091:
4089:
4088:
4083:
4071:
4069:
4068:
4063:
4051:
4049:
4048:
4043:
4028:
4026:
4025:
4020:
4000:
3998:
3997:
3996:
3974:
3951:
3946:
3945:
3920:
3918:
3917:
3912:
3900:
3898:
3897:
3892:
3890:
3889:
3870:
3868:
3867:
3862:
3860:
3858:
3857:
3848:
3847:
3846:
3822:
3821:
3811:
3788:
3787:
3775:
3774:
3769:
3765:
3760:
3749:
3720:
3718:
3717:
3712:
3700:
3698:
3697:
3692:
3680:
3678:
3677:
3672:
3670:
3669:
3650:
3648:
3647:
3642:
3640:
3638:
3637:
3628:
3627:
3626:
3602:
3601:
3591:
3568:
3567:
3555:
3554:
3549:
3545:
3540:
3529:
3519:
3518:
3496:
3494:
3493:
3488:
3486:
3482:
3481:
3477:
3473:
3468:
3457:
3441:
3440:
3434:
3430:
3429:
3428:
3419:
3414:
3403:
3379:
3375:
3370:
3359:
3350:
3349:
3337:
3336:
3331:
3327:
3322:
3311:
3301:
3300:
3281:
3279:
3278:
3273:
3256:
3248:
3247:
3222:
3220:
3219:
3214:
3188:
3187:
3158:
3156:
3155:
3150:
3148:
3144:
3137:
3132:
3121:
3104:
3103:
3088:
3083:
3072:
3059:
3055:
3050:
3039:
3030:
3029:
3017:
3016:
3011:
3007:
3002:
2991:
2981:
2980:
2958:
2956:
2955:
2950:
2948:
2944:
2943:
2938:
2927:
2922:
2921:
2912:
2907:
2896:
2876:
2875:
2866:
2861:
2850:
2837:
2836:
2824:
2823:
2818:
2814:
2809:
2798:
2788:
2787:
2768:
2766:
2765:
2760:
2749:
2747:
2746:
2741:
2739:
2738:
2716:
2714:
2713:
2708:
2706:
2705:
2689:
2687:
2686:
2681:
2679:
2678:
2656:
2654:
2653:
2648:
2640:
2639:
2614:
2612:
2611:
2606:
2595:
2594:
2576:
2571:
2560:
2555:
2554:
2535:
2533:
2532:
2527:
2525:
2524:
2505:
2503:
2502:
2497:
2485:
2483:
2482:
2477:
2453:
2451:
2450:
2445:
2385:
2384:
2380:
2375:
2374:
2370:
2357:
2355:
2354:
2349:
2329:
2328:
2305:
2303:
2302:
2297:
2277:
2276:
2251:
2249:
2248:
2243:
2161:
2159:
2158:
2153:
2139:
2131:
2071:
2069:
2068:
2063:
2009:
2007:
2006:
2001:
1959:
1957:
1956:
1951:
1872:
1870:
1869:
1864:
1849:
1847:
1846:
1841:
1791:
1789:
1788:
1783:
1767:
1765:
1764:
1759:
1757:
1756:
1738:
1737:
1706:
1705:
1632:
1630:
1629:
1624:
1575:
1573:
1572:
1567:
1521:
1519:
1518:
1513:
1490:
1488:
1487:
1482:
1450:
1448:
1447:
1442:
1419:
1282:4 4 4 1
1072:Base 5 numbering
1059:Norman H. Anning
1000:Williams version
995:
993:
992:
987:
949:
947:
946:
941:
894:
892:
891:
886:
849:
847:
846:
841:
839:
830:
821:
812:
797:
788:
779:
770:
764:
755:
745:
743:
742:
737:
735:
726:
717:
708:
678:
669:
659:
657:
656:
651:
355:David Singmaster
322:
321:
317:
312:
311:
307:
242:in the field of
228:
221:
210:
203:
199:
196:
190:
170:
169:
162:
155:
148:
144:
141:
135:
133:
92:
68:
60:
49:
27:
26:
19:
7158:
7157:
7153:
7152:
7151:
7149:
7148:
7147:
7113:
7112:
7094:
7087:at cut-the-knot
7013:
7008:
6999:
6995:
6976:
6973:
6972:
6935:
6931:
6893:
6890:
6889:
6873:
6870:
6869:
6832:
6828:
6796:
6793:
6792:
6776:
6773:
6772:
6753:
6750:
6749:
6711:
6707:
6705:
6702:
6701:
6699:
6695:
6686:
6682:
6673:
6669:
6647:
6643:
6638:
6634:
6629:
6625:
6620:
6616:
6611:
6607:
6602:
6598:
6587:
6583:
6578:
6574:
6564:
6560:
6552:
6548:
6543:
6539:
6534:
6530:
6526:Antonick (2013)
6525:
6521:
6514:
6495:
6480:
6476:Pleacher (2005)
6475:
6468:
6461:
6457:
6453:
6429:
6404:
6401:
6400:
6384:
6381:
6380:
6364:
6361:
6360:
6344:
6341:
6340:
6314:
6310:
6308:
6305:
6304:
6288:
6285:
6284:
6261:
6257:
6255:
6252:
6251:
6225:
6221:
6182:
6178:
6169:
6165:
6157:
6154:
6153:
6125:
6122:
6121:
6095:
6091:
6089:
6086:
6085:
6063:
6060:
6059:
6043:
6040:
6039:
6016:
6012:
6010:
6007:
6006:
5990:
5987:
5986:
5960:
5956:
5954:
5951:
5950:
5924:
5920:
5899:
5895:
5886:
5882:
5874:
5871:
5870:
5842:
5839:
5838:
5816:
5813:
5812:
5786:
5782:
5759:
5756:
5755:
5719:
5715:
5699:
5697:
5688:
5684:
5682:
5679:
5678:
5646:
5643:
5642:
5620:
5617:
5616:
5590:
5586:
5551:
5548:
5547:
5538:
5479:
5476:
5475:
5423:
5420:
5419:
5373:
5370:
5369:
5329:
5326:
5325:
5291:
5288:
5287:
5253:
5250:
5249:
5215:
5212:
5211:
5189:
5186:
5185:
5168:
5165:
5164:
5143:
5140:
5139:
5123:
5120:
5119:
5103:
5100:
5099:
5068:
5065:
5064:
5047:
5043:
5041:
5038:
5037:
5021:
5018:
5017:
4998:
4995:
4994:
4978:
4975:
4974:
4958:
4955:
4954:
4938:
4935:
4934:
4918:
4915:
4914:
4888:
4884:
4882:
4879:
4878:
4862:
4859:
4858:
4835:
4831:
4829:
4826:
4825:
4799:
4795:
4762:
4758:
4749:
4745:
4737:
4734:
4733:
4714:
4711:
4710:
4681:
4677:
4669:
4666:
4665:
4655:
4644:
4631:is even, where
4552:
4548:
4534:
4531:
4530:
4499:
4496:
4495:
4473:
4470:
4469:
4453:
4450:
4449:
4418:
4415:
4414:
4370:
4366:
4352:
4349:
4348:
4308:
4305:
4304:
4281:
4277:
4264:
4241:
4239:
4231:
4228:
4227:
4208:
4205:
4204:
4184:
4180:
4167:
4144:
4142:
4140:
4137:
4136:
4120:
4117:
4116:
4097:
4094:
4093:
4077:
4074:
4073:
4057:
4054:
4053:
4037:
4034:
4033:
3992:
3988:
3975:
3952:
3950:
3941:
3937:
3929:
3926:
3925:
3906:
3903:
3902:
3885:
3881:
3879:
3876:
3875:
3853:
3849:
3842:
3838:
3817:
3813:
3812:
3810:
3783:
3779:
3770:
3750:
3748:
3744:
3743:
3729:
3726:
3725:
3706:
3703:
3702:
3686:
3683:
3682:
3665:
3661:
3659:
3656:
3655:
3633:
3629:
3622:
3618:
3597:
3593:
3592:
3590:
3563:
3559:
3550:
3530:
3528:
3524:
3523:
3514:
3510:
3508:
3505:
3504:
3458:
3456:
3446:
3442:
3436:
3435:
3424:
3420:
3404:
3402:
3392:
3388:
3387:
3383:
3360:
3358:
3354:
3345:
3341:
3332:
3312:
3310:
3306:
3305:
3296:
3292:
3290:
3287:
3286:
3252:
3243:
3239:
3228:
3225:
3224:
3177:
3173:
3171:
3168:
3167:
3122:
3120:
3093:
3089:
3073:
3071:
3067:
3063:
3040:
3038:
3034:
3025:
3021:
3012:
2992:
2990:
2986:
2985:
2976:
2972:
2970:
2967:
2966:
2928:
2926:
2917:
2913:
2897:
2895:
2871:
2867:
2851:
2849:
2845:
2841:
2832:
2828:
2819:
2799:
2797:
2793:
2792:
2783:
2779:
2777:
2774:
2773:
2754:
2751:
2750:
2728:
2724:
2722:
2719:
2718:
2701:
2697:
2695:
2692:
2691:
2668:
2664:
2662:
2659:
2658:
2629:
2625:
2623:
2620:
2619:
2584:
2580:
2561:
2559:
2550:
2546:
2544:
2541:
2540:
2520:
2516:
2514:
2511:
2510:
2491:
2488:
2487:
2471:
2468:
2467:
2464:
2406:
2403:
2402:
2393:
2389:
2382:
2378:
2377:
2372:
2368:
2367:
2364:
2324:
2320:
2312:
2309:
2308:
2272:
2268:
2260:
2257:
2256:
2237:
2234:
2233:
2231:
2227:
2168:
2130:
2095:
2092:
2091:
2083: 1024, so 5
2021:
2018:
2017:
1971:
1968:
1967:
1882:
1879:
1878:
1858:
1855:
1854:
1808:
1805:
1804:
1798:
1777:
1774:
1773:
1752:
1751:
1733:
1729:
1720:
1719:
1701:
1697:
1684:
1683:
1681:
1678:
1677:
1668:
1664:
1656:
1652:
1648:
1644:
1600:
1597:
1596:
1543:
1540:
1539:
1533:
1498:
1495:
1494:
1458:
1455:
1454:
1415:
1404:
1401:
1400:
1347:
1307:
1301:
1297:
1292:
1288:
1283:
1281:
1277:
1273:
1269:
1265:
1261:
1254:
1250:
1246:
1241:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1202:
1200:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1165:
1161:
1157:
1153:
1148:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1091:
1087:
1083:
1079:
1074:
1051:
1018:
1002:
963:
960:
959:
911:
908:
907:
862:
859:
858:
828:
810:
786:
768:
753:
751:
748:
747:
724:
706:
667:
665:
662:
661:
636:
633:
632:
614:
602:
597:
590:
579:
572:
557:
543:
481:
477:
456:Clifton Fadiman
382:W.W. Rouse Ball
319:
315:
314:
309:
305:
304:
297:
264:
229:
218:
217:
216:
211:
200:
194:
191:
183:help improve it
180:
171:
167:
156:
145:
139:
136:
93:
91:
81:
69:
28:
24:
17:
12:
11:
5:
7156:
7146:
7145:
7140:
7135:
7130:
7125:
7111:
7110:
7101:
7093:
7092:External links
7090:
7089:
7088:
7076:
7067:
7064:
7056:
7042:Pappas, Theoni
7039:
7030:
7027:New York Times
7017:Antonick, Gary
7012:
7009:
7007:
7006:
6993:
6980:
6970:
6969:
6958:
6955:
6952:
6949:
6946:
6943:
6938:
6934:
6930:
6927:
6924:
6921:
6918:
6915:
6912:
6909:
6906:
6903:
6900:
6897:
6877:
6866:
6855:
6852:
6849:
6846:
6843:
6840:
6835:
6831:
6827:
6824:
6821:
6818:
6815:
6812:
6809:
6806:
6803:
6800:
6780:
6757:
6737:
6734:
6731:
6728:
6725:
6722:
6719:
6714:
6710:
6693:
6680:
6667:
6641:
6632:
6623:
6614:
6605:
6596:
6581:
6572:
6558:
6546:
6537:
6528:
6519:
6512:
6498:Martin Gardner
6478:
6466:
6454:
6452:
6449:
6448:
6447:
6442:
6436:
6428:
6425:
6408:
6388:
6368:
6348:
6328:
6325:
6322:
6317:
6313:
6292:
6272:
6269:
6264:
6260:
6248:
6247:
6234:
6231:
6228:
6224:
6220:
6217:
6214:
6211:
6208:
6205:
6202:
6199:
6196:
6193:
6190:
6185:
6181:
6177:
6172:
6168:
6164:
6161:
6135:
6132:
6129:
6109:
6106:
6103:
6098:
6094:
6073:
6070:
6067:
6047:
6027:
6024:
6019:
6015:
5994:
5974:
5971:
5968:
5963:
5959:
5947:
5946:
5933:
5930:
5927:
5923:
5919:
5916:
5913:
5910:
5907:
5902:
5898:
5894:
5889:
5885:
5881:
5878:
5852:
5849:
5846:
5826:
5823:
5820:
5809:
5808:
5795:
5792:
5789:
5785:
5781:
5778:
5775:
5772:
5769:
5766:
5763:
5748:
5747:
5736:
5733:
5728:
5725:
5722:
5718:
5712:
5708:
5705:
5702:
5696:
5691:
5687:
5656:
5653:
5650:
5630:
5627:
5624:
5613:
5612:
5599:
5596:
5593:
5589:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5561:
5558:
5555:
5537:
5534:
5531:
5530:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5473:
5469:
5468:
5457:
5454:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5417:
5413:
5412:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5367:
5363:
5362:
5351:
5348:
5345:
5342:
5339:
5336:
5333:
5323:
5319:
5318:
5307:
5304:
5301:
5298:
5295:
5285:
5281:
5280:
5269:
5266:
5263:
5260:
5257:
5247:
5243:
5242:
5231:
5228:
5225:
5222:
5219:
5209:
5205:
5204:
5193:
5183:
5172:
5147:
5127:
5107:
5072:
5050:
5046:
5025:
5002:
4982:
4962:
4942:
4922:
4902:
4899:
4896:
4891:
4887:
4866:
4846:
4843:
4838:
4834:
4822:
4821:
4808:
4805:
4802:
4798:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4765:
4761:
4757:
4752:
4748:
4744:
4741:
4718:
4698:
4695:
4692:
4689:
4684:
4680:
4676:
4673:
4653:
4642:
4593:
4592:
4581:
4576:
4572:
4569:
4566:
4563:
4560:
4555:
4551:
4547:
4544:
4541:
4538:
4515:
4512:
4509:
4506:
4503:
4483:
4480:
4477:
4457:
4446:
4445:
4434:
4431:
4428:
4425:
4422:
4402:
4399:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4373:
4369:
4365:
4362:
4359:
4356:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4301:
4300:
4284:
4280:
4276:
4273:
4270:
4267:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4238:
4235:
4212:
4187:
4183:
4179:
4176:
4173:
4170:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4124:
4101:
4081:
4061:
4041:
4030:
4029:
4018:
4015:
4012:
4009:
4006:
4003:
3995:
3991:
3987:
3984:
3981:
3978:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3949:
3944:
3940:
3936:
3933:
3910:
3888:
3884:
3872:
3871:
3856:
3852:
3845:
3841:
3837:
3834:
3831:
3828:
3825:
3820:
3816:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3786:
3782:
3778:
3773:
3768:
3763:
3759:
3756:
3753:
3747:
3742:
3739:
3736:
3733:
3710:
3690:
3668:
3664:
3652:
3651:
3636:
3632:
3625:
3621:
3617:
3614:
3611:
3608:
3605:
3600:
3596:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3566:
3562:
3558:
3553:
3548:
3543:
3539:
3536:
3533:
3527:
3522:
3517:
3513:
3498:
3497:
3485:
3480:
3476:
3471:
3467:
3464:
3461:
3455:
3452:
3449:
3445:
3439:
3433:
3427:
3423:
3417:
3413:
3410:
3407:
3401:
3398:
3395:
3391:
3386:
3382:
3378:
3373:
3369:
3366:
3363:
3357:
3353:
3348:
3344:
3340:
3335:
3330:
3325:
3321:
3318:
3315:
3309:
3304:
3299:
3295:
3271:
3268:
3265:
3262:
3259:
3255:
3251:
3246:
3242:
3238:
3235:
3232:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3186:
3183:
3180:
3176:
3160:
3159:
3147:
3143:
3140:
3135:
3131:
3128:
3125:
3119:
3116:
3113:
3110:
3107:
3102:
3099:
3096:
3092:
3086:
3082:
3079:
3076:
3070:
3066:
3062:
3058:
3053:
3049:
3046:
3043:
3037:
3033:
3028:
3024:
3020:
3015:
3010:
3005:
3001:
2998:
2995:
2989:
2984:
2979:
2975:
2960:
2959:
2947:
2941:
2937:
2934:
2931:
2925:
2920:
2916:
2910:
2906:
2903:
2900:
2894:
2891:
2888:
2885:
2882:
2879:
2874:
2870:
2864:
2860:
2857:
2854:
2848:
2844:
2840:
2835:
2831:
2827:
2822:
2817:
2812:
2808:
2805:
2802:
2796:
2791:
2786:
2782:
2769:times yields:
2758:
2737:
2734:
2731:
2727:
2704:
2700:
2677:
2674:
2671:
2667:
2646:
2643:
2638:
2635:
2632:
2628:
2616:
2615:
2604:
2601:
2598:
2593:
2590:
2587:
2583:
2579:
2574:
2570:
2567:
2564:
2558:
2553:
2549:
2523:
2519:
2495:
2475:
2463:
2460:
2456:
2455:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2391:
2387:
2363:
2360:
2359:
2358:
2347:
2344:
2341:
2338:
2335:
2332:
2327:
2323:
2319:
2316:
2306:
2295:
2292:
2289:
2286:
2283:
2280:
2275:
2271:
2267:
2264:
2241:
2229:
2225:
2224:So the pair (N
2167:
2164:
2163:
2162:
2151:
2148:
2145:
2142:
2137:
2134:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2073:
2072:
2061:
2056:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2011:
2010:
1999:
1994:
1990:
1987:
1984:
1981:
1978:
1975:
1961:
1960:
1949:
1944:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1862:
1851:
1850:
1839:
1834:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1797:
1794:
1781:
1770:
1769:
1755:
1750:
1747:
1744:
1741:
1736:
1732:
1728:
1725:
1722:
1721:
1718:
1715:
1712:
1709:
1704:
1700:
1696:
1693:
1690:
1689:
1687:
1666:
1662:
1654:
1650:
1646:
1642:
1635:
1634:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1578:
1577:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1532:
1529:
1524:
1523:
1511:
1508:
1505:
1502:
1492:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1452:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1418:
1414:
1411:
1408:
1346:
1343:
1335:
1334:
1331:
1328:
1325:
1322:
1315:
1314:
1306:
1303:
1299:
1295:
1290:
1286:
1279:
1275:
1271:
1267:
1263:
1259:
1257:
1252:
1248:
1244:
1240:4 4 1
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1207:
1201:4 1
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1168:
1163:
1159:
1155:
1151:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1094:
1089:
1085:
1081:
1077:
1073:
1070:
1050:
1047:
1017:
1014:
1001:
998:
985:
982:
979:
976:
973:
970:
967:
939:
936:
933:
930:
927:
924:
921:
918:
915:
896:
895:
884:
881:
878:
875:
872:
869:
866:
836:
833:
827:
824:
818:
815:
809:
806:
803:
800:
794:
791:
785:
782:
776:
773:
767:
761:
758:
732:
729:
723:
720:
714:
711:
705:
702:
699:
696:
693:
690:
687:
684:
681:
675:
672:
649:
646:
643:
640:
625:Martin Gardner
613:
610:
595:
588:
577:
570:
565:
564:
555:
551:
541:
505:
504:
499:
479:
478:
476:
473:
438:Martin Gardner
432:Horace Lorimer
424:
423:
419:
418:
413:
412:
407:
406:
359:Jacques Ozanam
296:
293:
276:
275:
272:
263:
260:
231:
230:
213:
212:
195:September 2022
174:
172:
165:
158:
157:
72:
70:
63:
58:
32:
31:
29:
22:
15:
9:
6:
4:
3:
2:
7155:
7144:
7141:
7139:
7136:
7134:
7131:
7129:
7126:
7124:
7121:
7120:
7118:
7109:
7105:
7102:
7099:
7096:
7095:
7086:
7085:
7080:
7077:
7074:
7073:
7068:
7065:
7063:
7062:
7057:
7055:
7051:
7047:
7043:
7040:
7037:
7036:
7031:
7028:
7024:
7023:
7018:
7015:
7014:
7003:
6997:
6978:
6953:
6950:
6947:
6941:
6936:
6932:
6928:
6922:
6919:
6916:
6913:
6910:
6907:
6904:
6898:
6895:
6875:
6867:
6850:
6847:
6844:
6838:
6833:
6829:
6825:
6819:
6816:
6813:
6810:
6807:
6801:
6798:
6778:
6770:
6769:
6755:
6735:
6732:
6729:
6726:
6723:
6720:
6717:
6712:
6708:
6697:
6690:
6684:
6677:
6671:
6663:
6659:
6655:
6652:(June 1912).
6651:
6645:
6636:
6627:
6618:
6609:
6600:
6594:
6590:
6585:
6576:
6569:
6568:
6562:
6556:July 27, 1962
6555:
6550:
6541:
6532:
6523:
6515:
6513:0-393-02023-1
6509:
6505:
6504:
6499:
6493:
6491:
6489:
6487:
6485:
6483:
6473:
6471:
6464:
6459:
6455:
6446:
6443:
6440:
6437:
6434:
6431:
6430:
6424:
6420:
6406:
6386:
6366:
6346:
6326:
6323:
6320:
6315:
6311:
6290:
6270:
6267:
6262:
6258:
6232:
6229:
6226:
6222:
6218:
6215:
6212:
6206:
6203:
6200:
6194:
6191:
6188:
6183:
6179:
6175:
6170:
6166:
6162:
6159:
6152:
6151:
6150:
6147:
6133:
6130:
6127:
6107:
6104:
6101:
6096:
6092:
6071:
6068:
6065:
6045:
6025:
6022:
6017:
6013:
5992:
5972:
5969:
5966:
5961:
5957:
5931:
5928:
5925:
5921:
5917:
5914:
5911:
5908:
5905:
5900:
5896:
5892:
5887:
5883:
5879:
5876:
5869:
5868:
5867:
5864:
5850:
5847:
5844:
5824:
5821:
5818:
5793:
5790:
5787:
5783:
5779:
5776:
5773:
5770:
5767:
5764:
5761:
5754:
5753:
5752:
5734:
5731:
5726:
5723:
5720:
5716:
5710:
5706:
5703:
5700:
5694:
5689:
5685:
5677:
5676:
5675:
5673:
5668:
5654:
5651:
5648:
5628:
5625:
5622:
5597:
5594:
5591:
5587:
5583:
5580:
5577:
5571:
5568:
5565:
5559:
5556:
5553:
5546:
5545:
5544:
5541:
5517:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5474:
5471:
5470:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5425:
5418:
5415:
5414:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5368:
5365:
5364:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5324:
5321:
5320:
5305:
5302:
5299:
5296:
5293:
5286:
5283:
5282:
5267:
5264:
5261:
5258:
5255:
5248:
5245:
5244:
5229:
5226:
5223:
5220:
5217:
5210:
5207:
5206:
5191:
5184:
5170:
5163:
5162:
5159:
5145:
5125:
5105:
5096:
5094:
5090:
5086:
5070:
5048:
5044:
5023:
5014:
5000:
4980:
4960:
4940:
4920:
4900:
4897:
4894:
4889:
4885:
4864:
4844:
4841:
4836:
4832:
4806:
4803:
4800:
4796:
4792:
4789:
4786:
4780:
4777:
4774:
4768:
4763:
4759:
4755:
4750:
4746:
4742:
4739:
4732:
4731:
4730:
4716:
4696:
4693:
4690:
4687:
4682:
4678:
4674:
4671:
4662:
4660:
4656:
4649:
4645:
4639:. Therefore
4638:
4634:
4630:
4626:
4622:
4618:
4614:
4610:
4606:
4602:
4598:
4579:
4574:
4570:
4567:
4564:
4561:
4558:
4553:
4545:
4542:
4539:
4529:
4528:
4527:
4510:
4507:
4504:
4481:
4478:
4475:
4455:
4432:
4429:
4426:
4423:
4420:
4400:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4371:
4363:
4360:
4357:
4347:
4346:
4345:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4282:
4274:
4271:
4268:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4236:
4233:
4226:
4225:
4224:
4210:
4185:
4177:
4174:
4171:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4122:
4113:
4099:
4079:
4059:
4039:
4013:
4010:
4007:
4001:
3993:
3985:
3982:
3979:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3947:
3942:
3938:
3934:
3931:
3924:
3923:
3922:
3908:
3886:
3882:
3854:
3850:
3843:
3835:
3832:
3829:
3823:
3818:
3814:
3807:
3801:
3798:
3795:
3789:
3784:
3780:
3776:
3771:
3766:
3761:
3757:
3754:
3751:
3745:
3740:
3737:
3734:
3731:
3724:
3723:
3722:
3708:
3688:
3666:
3662:
3634:
3630:
3623:
3615:
3612:
3609:
3603:
3598:
3594:
3587:
3581:
3578:
3575:
3569:
3564:
3560:
3556:
3551:
3546:
3541:
3537:
3534:
3531:
3525:
3520:
3515:
3511:
3503:
3502:
3501:
3483:
3478:
3469:
3465:
3462:
3459:
3450:
3447:
3443:
3431:
3425:
3415:
3411:
3408:
3405:
3396:
3393:
3389:
3384:
3380:
3376:
3371:
3367:
3364:
3361:
3355:
3351:
3346:
3342:
3338:
3333:
3328:
3323:
3319:
3316:
3313:
3307:
3302:
3297:
3293:
3285:
3284:
3283:
3266:
3263:
3260:
3253:
3244:
3240:
3236:
3233:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3184:
3181:
3178:
3174:
3165:
3145:
3141:
3138:
3133:
3129:
3126:
3123:
3117:
3114:
3111:
3108:
3105:
3100:
3097:
3094:
3084:
3080:
3077:
3074:
3064:
3060:
3056:
3051:
3047:
3044:
3041:
3035:
3031:
3026:
3022:
3018:
3013:
3008:
3003:
2999:
2996:
2993:
2987:
2982:
2977:
2973:
2965:
2964:
2963:
2945:
2939:
2935:
2932:
2929:
2923:
2918:
2908:
2904:
2901:
2898:
2889:
2886:
2883:
2880:
2877:
2872:
2862:
2858:
2855:
2852:
2842:
2838:
2833:
2829:
2825:
2820:
2815:
2810:
2806:
2803:
2800:
2794:
2789:
2784:
2780:
2772:
2771:
2770:
2756:
2735:
2732:
2729:
2725:
2702:
2698:
2675:
2669:
2665:
2644:
2641:
2636:
2630:
2626:
2599:
2596:
2591:
2588:
2585:
2581:
2572:
2568:
2565:
2562:
2556:
2551:
2547:
2539:
2538:
2537:
2521:
2517:
2507:
2493:
2473:
2459:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2401:
2400:
2399:
2397:
2386:, giving us x
2345:
2342:
2339:
2336:
2333:
2330:
2325:
2321:
2317:
2314:
2307:
2293:
2290:
2287:
2284:
2281:
2278:
2273:
2269:
2265:
2262:
2255:
2254:
2253:
2239:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2178:
2174:
2149:
2146:
2143:
2135:
2132:
2127:
2124:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2090:
2089:
2088:
2086:
2082:
2078:
2059:
2054:
2047:
2044:
2041:
2038:
2032:
2029:
2026:
2023:
2016:
2015:
2014:
1997:
1992:
1988:
1985:
1982:
1979:
1976:
1973:
1966:
1965:
1964:
1963:subtracting,
1947:
1942:
1935:
1932:
1929:
1926:
1923:
1917:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1887:
1884:
1877:
1876:
1875:
1860:
1837:
1832:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1803:
1802:
1801:
1793:
1779:
1748:
1745:
1742:
1739:
1734:
1730:
1726:
1723:
1716:
1713:
1710:
1707:
1702:
1698:
1694:
1691:
1685:
1676:
1675:
1674:
1672:
1660:
1640:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1595:
1594:
1593:
1590:
1587:
1583:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1538:
1537:
1536:
1528:
1509:
1506:
1503:
1500:
1493:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1453:
1435:
1432:
1429:
1423:
1420:
1416:
1412:
1409:
1406:
1399:
1398:
1397:
1393:
1390:
1388:
1384:
1380:
1377: 25 – 4.
1376:
1372:
1368:
1363:
1361:
1357:
1353:
1342:
1340:
1332:
1329:
1326:
1323:
1320:
1319:
1318:
1312:
1311:
1310:
1302:
1256:
1206:
1167:
1093:
1069:
1067:
1062:
1060:
1055:
1049:Blue coconuts
1046:
1043:
1039:
1035:
1031:
1027:
1022:
1016:Using a sieve
1013:
1006:
997:
983:
980:
977:
974:
971:
968:
965:
957:
953:
937:
934:
931:
928:
925:
922:
919:
916:
913:
905:
901:
882:
879:
876:
873:
870:
867:
864:
857:
856:
855:
853:
834:
831:
825:
822:
816:
813:
807:
804:
801:
792:
789:
783:
780:
774:
771:
759:
756:
730:
727:
721:
718:
712:
709:
703:
700:
697:
691:
688:
685:
682:
673:
670:
647:
644:
641:
638:
630:
626:
618:
609:
605:
600:
598:
591:
584:
580:
573:
563:
561:
552:
549:
547:
538:
537:
536:
534:
530:
526:
522:
518:
514:
510:
503:
500:
497:
494:
493:
492:
489:
486:
472:
469:
467:
463:
462:
457:
453:
449:
448:
443:
439:
435:
433:
429:
421:
420:
415:
414:
409:
408:
404:
403:
402:
400:
399:
394:
389:
387:
383:
379:
378:Lewis Carroll
374:
372:
368:
364:
360:
356:
351:
349:
348:
343:
338:
334:
330:
326:
302:
292:
290:
284:
280:
273:
269:
268:
267:
259:
258:collections.
257:
253:
249:
245:
241:
237:
227:
224:
209:
206:
198:
188:
184:
178:
175:This article
173:
164:
163:
154:
151:
143:
140:December 2019
132:
129:
125:
122:
118:
115:
111:
108:
104:
101: –
100:
96:
95:Find sources:
89:
85:
79:
78:
73:This article
71:
67:
62:
61:
56:
54:
47:
46:
41:
40:
35:
30:
21:
20:
7107:
7082:
7070:
7059:
7045:
7038:May 16, 2005
7033:
7026:
7020:
7001:
6996:
6696:
6683:
6675:
6670:
6661:
6657:
6644:
6635:
6626:
6617:
6608:
6599:
6588:
6584:
6575:
6566:
6561:
6549:
6540:
6531:
6522:
6502:
6458:
6421:
6249:
6148:
6005:is odd, and
5948:
5865:
5810:
5749:
5671:
5669:
5614:
5542:
5539:
5097:
5092:
5088:
5084:
5015:
4823:
4663:
4658:
4651:
4647:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4612:
4608:
4604:
4600:
4596:
4594:
4447:
4302:
4114:
4031:
3874:Solving for
3873:
3653:
3499:
3164:power series
3161:
2961:
2617:
2508:
2465:
2457:
2395:
2390:=–4776 and y
2365:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2176:
2172:
2169:
2084:
2080:
2076:
2074:
2012:
1962:
1852:
1799:
1771:
1670:
1661: 8404, F
1658:
1636:
1592:Reordering,
1591:
1585:
1581:
1579:
1534:
1525:
1394:
1391:
1386:
1382:
1378:
1374:
1370:
1366:
1364:
1359:
1355:
1351:
1348:
1338:
1336:
1316:
1308:
1284:
1242:
1203:
1149:
1075:
1063:
1056:
1052:
1041:
1037:
1033:
1029:
1025:
1023:
1019:
1011:
955:
954:and 1024 to
951:
897:
851:
628:
623:
606:
603:
593:
586:
582:
575:
568:
566:
559:
553:
545:
539:
532:
528:
520:
516:
508:
506:
501:
495:
490:
483:
470:
459:
445:
436:
427:
425:
396:
390:
385:
375:
362:
352:
345:
324:
298:
285:
281:
277:
265:
235:
234:
219:
201:
192:
176:
146:
137:
127:
120:
113:
106:
94:
82:Please help
77:verification
74:
50:
43:
37:
36:Please help
33:
4650:is odd and
4615:is odd and
2013:factoring,
496:ax + by = c
7117:Categories
7054:0933174659
6451:References
5087:= 1
5063:is 1, and
2171:to obtain
904:Paul Dirac
350:in 300CE.
110:newspapers
39:improve it
6951:−
6942:−
6929:⋅
6920:⋅
6908:−
6848:−
6839:−
6826:⋅
6817:⋅
6733:⋅
6721:−
6324:−
6219:⋅
6204:−
6195:⋅
6189:−
6176:⋅
6105:−
5970:−
5918:⋅
5906:−
5893:⋅
5780:⋅
5768:−
5732:−
5724:−
5704:−
5584:⋅
5569:−
5560:−
5503:⋅
5441:⋅
5391:⋅
5341:⋅
5303:⋅
5265:⋅
5227:⋅
4933:even and
4898:−
4793:⋅
4778:−
4769:−
4756:⋅
4694:⋅
4568:−
4565:≡
4559:⋅
4543:−
4479:−
4398:−
4389:⋅
4383:−
4377:⋅
4361:−
4329:±
4272:−
4258:⋅
4246:−
4175:−
4161:⋅
4149:−
4011:−
4002:−
3983:−
3969:⋅
3957:−
3948:⋅
3833:−
3824:−
3808:⋅
3799:−
3790:−
3777:⋅
3755:−
3735:⋅
3613:−
3604:−
3588:⋅
3579:−
3570:−
3557:⋅
3535:−
3463:−
3451:−
3409:−
3397:−
3381:⋅
3365:−
3352:−
3339:⋅
3317:−
3264:−
3237:−
3182:−
3127:−
3098:−
3078:−
3061:⋅
3045:−
3032:−
3019:⋅
2997:−
2933:−
2902:−
2856:−
2839:−
2826:⋅
2804:−
2733:−
2673:→
2642:≡
2634:→
2597:−
2589:−
2566:−
2433:⋅
2421:⋅
2415:−
2337:⋅
2285:⋅
2141:⇒
2122:⇒
2110:⋅
2033:⋅
1933:⋅
1927:−
1906:⋅
1900:−
1746:⋅
1714:⋅
1609:−
1507:−
1504:≡
1476:−
1467:≡
1433:−
1424:⋅
1410:≡
1270:4 4 0 s
935:−
920:−
558:+ t
544:+ t
475:Solutions
411:together.
289:algebraic
45:talk page
7143:Coconuts
7104:Coconuts
7044:(1993).
7019:(2013).
6500:(2001).
6427:See also
6303:odd and
4877:odd and
4448:Because
4413:, where
3223:sums to
531:divides
430:editor,
367:Lagrange
347:Elements
301:Mahāvīra
252:coconuts
7128:Puzzles
7081:(1996)
7011:Sources
4657:=–1 if
2381:⁄
2371:⁄
2133:3195904
1491:(mod 5)
1294:N=44441
1224:4 0 s
1185:4 0 s
567:where (
511:is the
333:Sun Tzu
323:of his
318:⁄
308:⁄
295:History
181:Please
124:scholar
7052:
7000:While
6971:where
6888:even,
6510:
6359:even,
6250:where
5949:where
5089:·
4824:where
4646:=1 if
4627:+1 if
4621:·
4619:
4611:–1 if
4605:·
4603:
3701:(i.e.
2618:where
2081:·
1772:where
1671:·
1669:
1659:·
1657:
1580:where
1375:·
1373:
1367:·
1356:·
1352:·
1339:·
1115:0 (N
562: a
560:·
548: b
546:·
507:where
353:Prof.
342:Euclid
248:monkey
126:
119:
112:
105:
97:
7100:video
6748:when
5672:after
4623:
4607:
4595:But (
2430:15625
2373:15625
2282:15625
2107:15625
1894:15625
1820:15625
1711:15625
1612:15625
1555:15625
1358:
1298:=3121
1289:and n
1119:) s
972:15621
902:(via
883:11529
874:15625
554:y = y
540:x = x
523:. By
313:, 132
238:is a
131:JSTOR
117:books
7050:ISBN
6868:for
6771:for
6689:here
6687:See
6664:(6).
6508:ISBN
6339:for
6283:for
6120:and
5837:and
5641:and
5615:For
5332:3121
5306:1024
4913:for
4857:for
4468:and
4052:and
3654:But
3282:so,
3162:The
2717:for
2442:3121
2436:2569
2424:8404
2418:4776
2383:4776
2369:1024
2340:2569
2334:1024
2294:3121
2288:2569
2175:and
2150:3121
2136:1024
2119:8404
2098:1024
2060:1024
1998:1024
1948:1024
1936:1024
1924:8404
1909:1024
1838:1024
1829:8404
1811:1024
1743:1024
1621:8404
1603:1024
1564:8404
1546:1024
984:1023
865:1024
519:and
428:Post
103:news
6038:if
5985:if
5518:728
5512:217
5506:134
5494:505
5488:440
5482:117
5456:801
5450:764
5432:537
5426:823
5400:936
5394:279
5382:275
5376:233
5350:625
5294:765
4575:mod
3921:),
2379:313
2346:204
2113:204
2055:mod
1993:mod
1989:212
1980:265
1943:mod
1833:mod
1768:(3)
1645:, y
1633:(2)
1576:(1)
1024:If
832:241
814:125
515:of
444:in
384:'s
361:'s
185:to
86:by
7119::
6662:12
6660:.
6656:.
6481:^
6469:^
6084:,
5472:8
5416:7
5366:6
5344:15
5322:5
5284:4
5268:81
5256:25
5246:3
5218:11
5208:2
4223::
4112:.
4072:.
3901:(=
2506:.
2228:,F
2030:53
1903:15
1665:=y
1653:=x
1300:10
1162:=s
1147:)
1139:(S
1088:,s
1080:,n
1061:.
996:.
835:16
817:16
772:25
710:25
48:.
7002:N
6979:k
6957:)
6954:1
6948:m
6945:(
6937:m
6933:m
6926:)
6923:k
6917:m
6914:+
6911:1
6905:m
6902:(
6899:=
6896:N
6876:m
6854:)
6851:1
6845:m
6842:(
6834:m
6830:m
6823:)
6820:k
6814:m
6811:+
6808:1
6805:(
6802:=
6799:N
6779:m
6756:m
6736:m
6730:1
6727:+
6724:1
6718:=
6713:0
6709:r
6676:k
6589:d
6516:.
6407:c
6387:k
6367:c
6347:m
6327:1
6321:=
6316:0
6312:r
6291:m
6271:1
6268:=
6263:0
6259:r
6233:1
6230:+
6227:m
6223:m
6216:k
6213:+
6210:)
6207:1
6201:m
6198:(
6192:c
6184:m
6180:m
6171:0
6167:r
6163:=
6160:N
6134:1
6131:=
6128:k
6108:1
6102:=
6097:0
6093:r
6072:3
6069:=
6066:m
6046:m
6026:1
6023:=
6018:0
6014:r
5993:m
5973:1
5967:=
5962:0
5958:r
5932:1
5929:+
5926:m
5922:m
5915:k
5912:+
5909:m
5901:m
5897:m
5888:0
5884:r
5880:=
5877:N
5851:1
5848:=
5845:k
5825:4
5822:=
5819:m
5794:1
5791:+
5788:m
5784:m
5777:k
5774:+
5771:m
5765:=
5762:N
5735:1
5727:1
5721:i
5717:n
5711:m
5707:1
5701:m
5695:=
5690:i
5686:n
5655:1
5652:=
5649:k
5629:5
5626:=
5623:m
5598:1
5595:+
5592:m
5588:m
5581:k
5578:+
5575:)
5572:1
5566:m
5563:(
5557:=
5554:N
5515:,
5509:,
5500:k
5497:+
5491:,
5485:,
5453:,
5447:,
5444:5
5438:k
5435:+
5429:,
5397:,
5388:k
5385:+
5379:,
5347:,
5338:k
5335:+
5300:k
5297:+
5262:k
5259:+
5230:8
5224:k
5221:+
5192:N
5171:m
5146:k
5138:(
5126:m
5106:N
5093:k
5085:N
5071:k
5049:0
5045:r
5024:m
5001:N
4981:k
4961:m
4941:k
4921:m
4901:1
4895:=
4890:0
4886:r
4865:m
4845:1
4842:=
4837:0
4833:r
4807:1
4804:+
4801:m
4797:m
4790:k
4787:+
4784:)
4781:1
4775:m
4772:(
4764:m
4760:m
4751:0
4747:r
4743:=
4740:N
4717:r
4697:m
4691:k
4688:+
4683:0
4679:r
4675:=
4672:r
4659:m
4654:0
4652:r
4648:m
4643:0
4641:r
4637:m
4633:Z
4629:m
4625:m
4617:Z
4613:m
4609:m
4601:Z
4597:m
4580:m
4571:1
4562:r
4554:m
4550:)
4546:1
4540:m
4537:(
4514:)
4511:s
4508:,
4505:r
4502:(
4482:1
4476:m
4456:m
4433:F
4430:+
4427:1
4424:=
4421:s
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981:=
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969:=
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956:F
952:N
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932:=
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880:+
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202:(
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193:(
179:.
153:)
147:(
142:)
138:(
128:·
121:·
114:·
107:·
80:.
55:)
51:(
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