3741:
215:
693:
191:), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the integer being factored increases, the number of operations required to perform the factorization on any computer increases drastically.
1945:
263:
by repeated application of this algorithm. The situation is more complicated with special-purpose factorization algorithms, whose benefits may not be realized as well or even at all with the factors produced during decomposition. For example, if
2211:
Kleinjung, Thorsten; Aoki, Kazumaro; Franke, Jens; Lenstra, Arjen K.; Thomé, Emmanuel; Bos, Joppe W.; Gaudry, Pierrick; Kruppa, Alexander; Montgomery, Peter L.; Osvik, Dag Arne; te Riele, Herman J. J.; Timofeev, Andrey; Zimmermann, Paul (2010).
537:
979:
A special-purpose factoring algorithm's running time depends on the properties of the number to be factored or on one of its unknown factors: size, special form, etc. The parameters which determine the running time vary among algorithms.
991:
algorithms, whose running time depends on the size of smallest prime factor. Given an integer of unknown form, these methods are usually applied before general-purpose methods to remove small factors. For example, naive
1305:
3612:
2759:
Chapter 5: Exponential
Factoring Algorithms, pp. 191â226. Chapter 6: Subexponential Factoring Algorithms, pp. 227â284. Section 7.4: Elliptic curve method, pp. 301â313.
1818:
125:. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient. A prime factorization algorithm typically involves
382:. While these are easily recognized as composite and prime respectively, Fermat's method will take much longer to factor the composite number because the starting value of
2909:
82:
is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example
688:{\displaystyle \exp \left(\left(\left({\tfrac {8}{3}}\right)^{\frac {2}{3}}+o(1)\right)\left(\log n\right)^{\frac {1}{3}}\left(\log \log n\right)^{\frac {2}{3}}\right).}
3616:
2193:
805:
785:
3124:
434:) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.
430:
In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore
Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (
2902:
2390:
Vandersypen, Lieven M. K.; et al. (2001). "Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance".
3157:
2063:
831:, meaning that both "yes" and "no" answers can be verified in polynomial time. An answer of "yes" can be certified by exhibiting a factorization
2062:
To obtain an algorithm for factoring any positive integer, it is necessary to add a few steps to this algorithm such as trial division, and the
4102:
2107:
441:, an 829-bit number with 250 decimal digits, in February 2020. The total computation time was roughly 2700 core-years of computing using Intel
36:
1166:
445:
6130 at 2.1 GHz. Like all recent factorization records, this factorization was completed with a highly optimized implementation of the
1094:
algorithm, has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor
179:
Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are
4097:
2993:
2957:
2895:
2689:
898:
guarantees that there is only one possible string of increasing primes that will be accepted, which shows that the problem is in both
4092:
3684:
3481:
3117:
1342:
The
SchnorrâSeysenâLenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance to have expected running time
194:
Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problemâfor example, the
494:. Neither the existence nor non-existence of such algorithms has been proved, but it is generally suspected that they do not exist.
442:
2988:
1020:
958:
that can test primality very quickly in practice if one is willing to accept a vanishingly small possibility of error. The ease of
2221:
Advances in
Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings
3349:
1132:
2830:
2801:
2568:
2551:
2496:
2367:
2309:
2264:
2172:
3291:
3110:
1046:
3220:
3397:
20:
3730:
2918:
2780:
2752:
2197:
895:
233:
141:
187:
long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by
3927:
3344:
3306:
3281:
3195:
1360:
1111:
1051:
1024:
188:
3486:
3377:
3296:
3286:
3225:
3188:
2767:
1106:
1056:
3740:
3314:
3162:
3065:
3035:
1331:
427:
whose factors are of similar size. For this reason, these are the integers used in cryptographic applications.
3045:
3677:
3567:
2952:
2340:
Buhler, J. P.; Lenstra, H. W. Jr.; Pomerance, Carl (1993). "Factoring integers with the number field sieve".
1940:{\displaystyle \left(\prod _{x\in X_{}}x^{r(x)}\right).\left(\prod _{q\in P_{\Delta }}f_{q}^{t(q)}\right)=1.}
1035:
412:
2454:
3562:
3529:
3491:
3392:
3443:
3081:
2096:
1416:
is an odd positive integer greater than a certain constant. In this factoring algorithm the discriminant
153:
91:
3881:
3917:
3608:
3598:
3557:
3333:
3327:
3301:
3172:
3019:
2983:
2962:
1127:
1061:
1010:
739:
516:
446:
512:
259:
Given a general algorithm for integer factorization, any integer can be factored into its constituent
3902:
3593:
3534:
3060:
1505:
1066:
2978:
2866:
3670:
3496:
3369:
3215:
3167:
2117:
256:. If composite, however, the polynomial time tests give no insight into how to obtain the factors.
4046:
183:, the product of two prime numbers. When they are both large, for instance more than two thousand
4056:
3922:
3846:
3511:
3086:
2823:
2075:
1627:
169:
3402:
2543:
2536:
1591:
The relation that will be used is a relation between the product of powers that is equal to the
58:
of integers. Every positive integer greater than 1 is either the product of two or more integer
4107:
3907:
3866:
3622:
3572:
3552:
2787:
1157:
In number theory, there are many integer factoring algorithms that heuristically have expected
909:
The problem is suspected to be outside all three of the complexity classes P, NP-complete, and
19:"Prime decomposition" redirects here. For the prime decomposition theorem for 3-manifolds, see
2865:, Neeraj Kayal, Nitin Saxena, "PRIMES is in P." Annals of Mathematics 160(2): 781â793 (2004).
2511:
2291:
2246:
140:
is known. However, it has not been proven that such an algorithm does not exist. The presumed
3836:
3632:
3248:
3177:
2947:
2937:
2652:
1319:
1099:
55:
4010:
3912:
3627:
3519:
3501:
3476:
3438:
3182:
3055:
2718:
2637:
2592:
2506:
2411:
2319:
2274:
955:
738:-bit number inputs. In 2001, Shor's algorithm was implemented for the first time, by using
423:-bit numbers, the most difficult to factor in practice using existing algorithms are those
8:
4071:
4066:
3861:
3856:
3841:
3780:
3637:
3603:
3524:
3428:
3387:
3382:
3359:
3263:
2814:
2152:
2101:
1592:
1145:
1005:
713:
3049:
3039:
2872:
2852:
Richard P. Brent, "Recent
Progress and Prospects for Integer Factorisation Algorithms",
2415:
3995:
3990:
3951:
3871:
3851:
3468:
3415:
3412:
3253:
3202:
3152:
2612:"A probabilistic factorization algorithm with quadratic forms of negative discriminant"
2435:
2401:
2323:
1330:
proposed by
Schnorr, Seysen, and Lenstra, which they proved only assuming the unproved
951:
885:
790:
770:
253:
149:
3662:
3209:
2709:
2671:
2628:
2611:
932:
a prime number?") appears to be much easier than the problem of specifying factors of
4031:
3971:
3588:
3544:
3258:
3235:
2887:
2826:
2797:
2776:
2748:
2584:
2547:
2492:
2480:
2427:
2363:
2327:
2305:
2260:
2168:
2130:
1352:
1086:
705:
2849:â SIQS and NFS â has helped complete some of the largest public factorizations known
2038:
is found. In order to prevent useless ambiguous forms from generating, build up the
1980:
of order dividing 2 to obtain a coprime factorization of the largest odd divisor of
1351:
by replacing the GRH assumption with the use of multipliers. The algorithm uses the
4061:
4036:
3956:
3942:
3876:
3760:
3720:
3433:
3014:
2862:
2736:
2704:
2667:
2623:
2580:
2439:
2419:
2392:
2355:
2347:
2297:
2252:
2242:
2224:
2160:
2122:
1473:
750:
746:
173:
161:
133:
63:
51:
2596:
2164:
515:. As of 2022, the algorithm with best theoretical asymptotic running time is the
4041:
3966:
3960:
3897:
3795:
3785:
3715:
3423:
3322:
2714:
2633:
2588:
2502:
2315:
2270:
2228:
2223:. Lecture Notes in Computer Science. Vol. 6223. Springer. pp. 333â350.
1323:
1158:
1117:
1014:
963:
914:
899:
824:
462:
438:
249:
199:
4051:
4005:
3831:
3815:
3805:
3775:
3453:
3354:
3339:
3243:
3144:
3091:
3009:
2819:
2740:
2527:
2476:
2468:
2039:
1356:
1311:
1122:
1000:
993:
959:
910:
479:
289:
245:
165:
126:
102:
33:
Can integer factorization be solved in polynomial time on a classical computer?
3102:
2256:
938:. The composite/prime problem can be solved in polynomial time (in the number
4086:
4000:
3800:
3790:
3770:
3448:
3133:
2472:
2301:
2287:
2112:
1440:
241:
2034:
then stop, otherwise find another ambiguous form until the factorization of
214:
25:
4015:
3932:
3810:
3755:
3725:
3458:
2762:
2431:
260:
237:
145:
67:
2857:
2846:
2653:"Fast and rigorous factorization under the generalized Riemann hypothesis"
2346:. Lecture Notes in Mathematics. Vol. 1554. Springer. pp. 50â94.
1604:. These relations will be used to construct a so-called ambiguous form of
198:. An algorithm that efficiently factors an arbitrary integer would render
2942:
2484:
2406:
966:
algorithm, as it is necessary to find large prime numbers to start with.
195:
157:
118:
43:
2791:
2455:"Computational Complexity Blog: Complexity Class of the Week: Factoring"
872:. An answer of "no" can be certified by exhibiting the factorization of
2881:
2531:
2491:, Princeton, New Jersey: Princeton University Press, pp. 575â604,
2351:
2213:
1622:
of order dividing 2. By calculating the corresponding factorization of
1315:
1095:
709:
424:
203:
2248:
The Proof is in the
Pudding: The Changing Nature of Mathematical Proof
2194:"[Cado-nfs-discuss] 795-bit factoring and discrete logarithms"
1300:{\displaystyle L_{n}\left=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}}
698:
For current computers, GNFS is the best published algorithm for large
3765:
3136:
2359:
2341:
1456:
can be restricted to a small set to guarantee the smoothness result.
983:
An important subclass of special-purpose factoring algorithms is the
458:
180:
137:
2423:
90:; the result is always unique up to the order of the factors by the
2735:
1630:, this ambiguous form provides the complete prime factorization of
712:
discovered an algorithm in 1994 that solves it in polynomial time.
3710:
2078:
as it makes random choices. Its expected running time is at most
431:
101:
using mental or pen-and-paper arithmetic, the simplest method is
59:
2569:"Refined analysis and improvements on some factoring algorithms"
1435:
is some positive multiplier. The algorithm expects that for one
298:
divisions to find the next factor. As a contrasting example, if
86:. Continuing this process until every factor is prime is called
3981:
2526:
1098:. Most general-purpose factoring algorithms are based on the
828:
84:
60 = 3 · 20 = 3 · (5 · 4)
749:
such as P, NP, and co-NP, the problem has to be stated as a
2210:
2133:â a way of writing a number as a sum of positive integers.
1550:
a sequence of relations between the set of generators and
2783:. Section 4.5.4: Factoring into Primes, pp. 379â417.
903:
184:
292:
will quickly produce the factors 3 and 19 but will take
144:
of this problem is important for the algorithms used in
3692:
3653:
indicate that algorithm is for numbers of special forms
2104:
for generating random numbers with their factorizations
1077:
A general-purpose factoring algorithm, also known as a
105:: checking if the number is divisible by prime numbers
2917:
2786:
2339:
2296:, Cambridge: Cambridge University Press, p. 230,
2155:, in van Tilborg, Henk C. A.; Jajodia, Sushil (eds.),
1186:
563:
132:
When the numbers are sufficiently large, no efficient
2522:
2520:
1821:
1169:
793:
773:
540:
2683:
2681:
1665:
is the negative discriminant of some quadratic form.
497:
There are published algorithms that are faster than
164:
have been brought to bear on the problem, including
27:
1403:
742:techniques on molecules that provide seven qubits.
461:has been published that can factor all integers in
2535:
2517:
2467:
2030:If the ambiguous form provides a factorization of
1939:
1299:
799:
779:
687:
2796:. Providence, RI: American Mathematical Society.
2678:
4084:
2687:
1452:. Lenstra and Pomerance show that the choice of
320:, Fermat's factorization method will begin with
3132:
2882:Dario Alpern's Integer factorization calculator
2690:"A Rigorous Time Bound for Factoring Integers"
2108:Canonical representation of a positive integer
519:(GNFS), first published in 1993, running on a
3678:
3118:
2903:
2688:Lenstra, H. W.; Pomerance, Carl (July 1992).
2475:(2008), "IV.20 Computational Complexity", in
2452:
2159:, Boston, MA: Springer US, pp. 611â618,
66:, or it is not, in which case it is called a
62:greater than 1, in which case it is called a
2697:Journal of the American Mathematical Society
2644:
2560:
2542:. Key College Publishing/Springer. pp.
1785:Collect a sequence of relations between set
1400:in which those integers are relative prime.
1318:. Some examples of those algorithms are the
437:The largest such semiprime yet factored was
248:whether the integer is prime can be done in
37:(more unsolved problems in computer science)
2603:
2389:
1139:
926:a composite number?" (or equivalently: "Is
884:; one can verify their primality using the
406:
3685:
3671:
3125:
3111:
2910:
2896:
2745:Prime Numbers: A Computational Perspective
2286:
2708:
2627:
2405:
2343:The development of the number field sieve
2157:Encyclopedia of Cryptography and Security
1152:
2884:â A web app for factoring large integers
2214:"Factorization of a 768-Bit RSA Modulus"
2069:
1337:
1021:Algebraic-group factorization algorithms
213:
16:Decomposition of a number into a product
2775:, Third Edition. Addison-Wesley, 1997.
2650:
2566:
2538:A Course in Computational Number Theory
2150:
1634:. This algorithm has these main steps:
969:
4085:
2811:
2609:
2489:The Princeton Companion to Mathematics
2241:
920:In contrast, the decision problem "Is
913:. It is therefore a candidate for the
878:into distinct primes, all larger than
236:, every positive integer has a unique
209:
4103:Unsolved problems in computer science
3666:
3106:
2891:
2204:
1047:Lenstra elliptic curve factorization
127:testing whether each factor is prime
28:Unsolved problem in computer science
3693:Divisibility-based sets of integers
2251:, New York: Springer, p. 203,
1133:Shanks's square forms factorization
888:, and then multiply them to obtain
13:
4098:Computational hardness assumptions
2919:Computational hardness assumptions
1895:
1384:is the set of triples of integers
1072:
1013:, which has two common flavors to
974:
704:(more than about 400 bits). For a
452:
21:Prime decomposition of 3-manifolds
14:
4119:
3731:Fundamental theorem of arithmetic
3334:Special number field sieve (SNFS)
3328:General number field sieve (GNFS)
2840:
2710:10.1090/S0894-0347-1992-1137100-0
2629:10.1090/S0025-5718-1987-0878705-X
954:. In addition, there are several
896:fundamental theorem of arithmetic
234:fundamental theorem of arithmetic
4093:Integer factorization algorithms
3739:
2958:Decisional composite residuosity
1404:SchnorrâSeysenâLenstra algorithm
1326:. Another such algorithm is the
1112:Continued fraction factorization
1017:: one by Floyd and one by Brent.
902:and co-UP. It is known to be in
2768:The Art of Computer Programming
2461:
2446:
2383:
2333:
2280:
2235:
2186:
2144:
1921:
1915:
1861:
1855:
1641:be the number to be factored.
1332:generalized Riemann hypothesis
1290:
1272:
1269:
1257:
1252:
1249:
1243:
1231:
1215:
1209:
601:
595:
74:is a composite number because
1:
2812:Warren, Henry S. Jr. (2013).
2729:
2672:10.1016/S1385-7258(88)80022-2
2165:10.1007/978-1-4419-5906-5_455
2074:The algorithm as stated is a
1412:that will be factored, where
1052:Fermat's factorization method
906:because of Shor's algorithm.
465:, that is, that can factor a
449:run on hundreds of machines.
413:Integer factorization records
318:13729 Ă 1372933 = 18848997157
304:is the product of the primes
189:Fermat's factorization method
129:each time a factor is found.
97:To factorize a small integer
3292:Lenstra elliptic curve (ECM)
2994:Computational DiffieâHellman
2854:Computing and Combinatorics"
2585:10.1016/0196-6774(82)90012-8
2453:Lance Fortnow (2002-09-13).
2229:10.1007/978-3-642-14623-7_18
1951:Construct an ambiguous form
1328:class group relations method
1107:Dixon's factorization method
1057:Euler's factorization method
7:
3082:Exponential time hypothesis
2097:Aurifeuillean factorization
2090:
1650:be a negative integer with
1504:. By constructing a set of
1420:is chosen as a multiple of
996:is a Category 1 algorithm.
813:have a factor smaller than
240:. (By convention, 1 is the
92:prime factorization theorem
10:
4126:
3599:Exponentiation by squaring
3282:Continued fraction (CFRAC)
2616:Mathematics of Computation
2567:Schnorr, Claus P. (1982).
2151:Lenstra, Arjen K. (2011),
1728:be a random prime form of
1561:are produced. The size of
1128:General number field sieve
1062:Special number field sieve
823:It is known to be in both
767:For every natural numbers
517:general number field sieve
447:general number field sieve
410:
50:is the decomposition of a
18:
4024:
3980:
3941:
3928:Superior highly composite
3890:
3824:
3748:
3737:
3698:
3646:
3581:
3543:
3510:
3467:
3411:
3368:
3272:
3234:
3143:
3092:Planted clique conjecture
3074:
3061:Ring learning with errors
3028:
3002:
2989:Decisional DiffieâHellman
2971:
2925:
2660:Indagationes Mathematicae
2651:Lenstra, Arjen K (1988).
2257:10.1007/978-0-387-48744-1
1613:, which is an element of
1067:Difference of two squares
962:is a crucial part of the
333:which immediately yields
150:RSA public-key encryption
3825:Constrained divisor sums
2773:Seminumerical Algorithms
2302:10.1017/CBO9780511804090
2293:Computational complexity
2137:
2118:Multiplicative partition
1140:Other notable algorithms
956:probabilistic algorithms
407:Current state of the art
3512:Greatest common divisor
3087:Unique games conjecture
3036:Shortest vector problem
3010:External DiffieâHellman
2875:MathWorld Headline News
2867:August 2005 version PDF
2856:, 2000, pp. 3â22.
2824:Pearson Education, Inc.
2610:Seysen, Martin (1987).
2219:. In Rabin, Tal (ed.).
2076:probabilistic algorithm
1148:, for quantum computers
1011:Pollard's rho algorithm
762:(Integer factorization)
745:In order to talk about
399:is a factor of 10 from
288:are very large primes,
218:Prime decomposition of
206:cryptography insecure.
170:algebraic number theory
117:, and so on, up to the
3623:Modular exponentiation
3066:Short integer solution
3046:Closest vector problem
2788:Samuel S. Wagstaff Jr.
2599:on September 24, 2017.
2290:; Barak, Boaz (2009),
1941:
1769:Find a generating set
1468:the set of all primes
1301:
1153:Heuristic running time
801:
781:
689:
360:and hence the factors
252:, for example, by the
229:
136:integer factorization
76:15 = 3 · 5
3706:Integer factorization
3350:Shanks's square forms
3274:Integer factorization
3249:Sieve of Eratosthenes
2953:Quadratic residuosity
2933:Integer factorization
2573:Journal of Algorithms
2070:Expected running time
1942:
1338:Rigorous running time
1320:elliptic curve method
1302:
1100:congruence of squares
1015:identify group cycles
802:
782:
690:
217:
154:RSA digital signature
48:integer factorization
3628:Montgomery reduction
3502:Function field sieve
3477:Baby-step giant-step
3323:Quadratic sieve (QS)
3056:Learning with errors
2793:The Joy of Factoring
2510:. See in particular
1819:
1661:is a multiplier and
1167:
970:Factoring algorithms
791:
771:
538:
3918:Colossally abundant
3749:Factorization forms
3638:Trachtenberg system
3604:Integer square root
3545:Modular square root
3264:Wheel factorization
3216:Quadratic Frobenius
3196:LucasâLehmerâRiesel
2873:âRSA-640 Factoredâ
2871:Eric W. Weisstein,
2416:2001Natur.414..883V
2153:"Integer Factoring"
1967:that is an element
1925:
1439:there exist enough
1355:of positive binary
1006:Wheel factorization
765: —
238:prime factorization
210:Prime decomposition
88:prime factorization
3903:Primitive abundant
3891:With many divisors
3530:Extended Euclidean
3469:Discrete logarithm
3398:SchönhageâStrassen
3254:Sieve of Pritchard
2979:Discrete logarithm
2963:Higher residuosity
2877:, November 8, 2005
2481:Barrow-Green, June
2352:10.1007/BFb0091539
1937:
1902:
1901:
1846:
1579:for some constant
1565:can be bounded by
1297:
1195:
1023:, among which are
952:AKS primality test
917:complexity class.
886:AKS primality test
797:
777:
759:
747:complexity classes
685:
572:
488:for some constant
254:AKS primality test
230:
4080:
4079:
3660:
3659:
3259:Sieve of Sundaram
3100:
3099:
3075:Non-cryptographic
2832:978-0-321-84268-8
2803:978-1-4704-1048-3
2553:978-1-930190-10-8
2498:978-0-691-11880-2
2400:(6866): 883â887.
2369:978-3-540-57013-4
2311:978-0-521-42426-4
2266:978-0-387-48908-7
2243:Krantz, Steven G.
2174:978-1-4419-5905-8
2131:Integer partition
1879:
1827:
1408:Given an integer
1293:
1194:
960:primality testing
800:{\displaystyle k}
780:{\displaystyle n}
757:
674:
637:
586:
571:
505:for all positive
499:O((1 +
174:quantum computing
4115:
4057:Harmonic divisor
3943:Aliquot sequence
3923:Highly composite
3847:Multiply perfect
3743:
3721:Divisor function
3687:
3680:
3673:
3664:
3663:
3609:Integer relation
3582:Other algorithms
3487:Pollard kangaroo
3378:Ancient Egyptian
3236:Prime-generating
3221:SolovayâStrassen
3134:Number-theoretic
3127:
3120:
3113:
3104:
3103:
3015:Sub-group hiding
2926:Number theoretic
2912:
2905:
2898:
2889:
2888:
2863:Manindra Agrawal
2836:
2815:Hacker's Delight
2807:
2758:
2737:Richard Crandall
2723:
2722:
2712:
2694:
2685:
2676:
2675:
2657:
2648:
2642:
2641:
2631:
2622:(178): 757â780.
2607:
2601:
2600:
2595:. Archived from
2564:
2558:
2557:
2541:
2524:
2515:
2509:
2465:
2459:
2458:
2450:
2444:
2443:
2409:
2407:quant-ph/0112176
2387:
2381:
2380:
2378:
2376:
2337:
2331:
2330:
2284:
2278:
2277:
2239:
2233:
2232:
2218:
2208:
2202:
2201:
2196:. Archived from
2190:
2184:
2183:
2182:
2181:
2148:
2125:
2102:Bach's algorithm
2086:
2056:
2049:
2037:
2033:
2026:
2006:
1990:
1983:
1979:
1966:
1946:
1944:
1943:
1938:
1930:
1926:
1924:
1910:
1900:
1899:
1898:
1870:
1866:
1865:
1864:
1845:
1844:
1843:
1812:
1788:
1781:
1772:
1765:
1763:
1760:
1759:
1757:
1756:
1751:
1748:
1741:
1736:
1727:
1713:
1703:
1671:
1664:
1660:
1656:
1649:
1640:
1633:
1626:and by taking a
1625:
1621:
1612:
1603:
1587:
1578:
1576:
1564:
1560:
1549:
1540:
1536:
1527:
1517:and prime forms
1516:
1503:
1501:
1498:
1497:
1495:
1494:
1489:
1486:
1479:
1474:Kronecker symbol
1471:
1467:
1455:
1451:
1438:
1434:
1430:
1423:
1419:
1415:
1411:
1399:
1383:
1374:
1365:
1350:
1306:
1304:
1303:
1298:
1296:
1295:
1294:
1256:
1222:
1218:
1196:
1187:
1179:
1178:
1146:Shor's algorithm
1042:
1031:
949:
943:
937:
931:
925:
893:
883:
877:
871:
861:
859:
857:
856:
851:
848:
818:
812:
806:
804:
803:
798:
786:
784:
783:
778:
766:
763:
758:Decision problem
751:decision problem
737:
731:
723:
714:Shor's algorithm
706:quantum computer
703:
694:
692:
691:
686:
681:
677:
676:
675:
667:
665:
661:
639:
638:
630:
628:
624:
608:
604:
588:
587:
579:
577:
573:
564:
530:
524:
510:
504:
493:
487:
476:
470:
422:
402:
398:
392:
390:
389:
381:
370:
359:
357:
356:
350:
349:
332:
330:
329:
319:
315:
311:
307:
303:
297:
287:
277:
228:
224:
162:computer science
156:. Many areas of
124:
116:
112:
108:
100:
85:
81:
77:
73:
64:composite number
52:positive integer
29:
4125:
4124:
4118:
4117:
4116:
4114:
4113:
4112:
4083:
4082:
4081:
4076:
4020:
3976:
3937:
3908:Highly abundant
3886:
3867:Unitary perfect
3820:
3744:
3735:
3716:Unitary divisor
3694:
3691:
3661:
3656:
3642:
3577:
3539:
3506:
3463:
3407:
3364:
3268:
3230:
3203:Proth's theorem
3145:Primality tests
3139:
3131:
3101:
3096:
3070:
3024:
3020:Decision linear
2998:
2972:Group theoretic
2967:
2921:
2916:
2843:
2833:
2804:
2755:
2732:
2727:
2726:
2692:
2686:
2679:
2655:
2649:
2645:
2608:
2604:
2565:
2561:
2554:
2525:
2518:
2499:
2477:Gowers, Timothy
2469:Goldreich, Oded
2466:
2462:
2451:
2447:
2424:10.1038/414883a
2388:
2384:
2374:
2372:
2370:
2338:
2334:
2312:
2285:
2281:
2267:
2240:
2236:
2216:
2209:
2205:
2192:
2191:
2187:
2179:
2177:
2175:
2149:
2145:
2140:
2126:-adic valuation
2123:
2093:
2084:
2079:
2072:
2064:Jacobi sum test
2060:
2051:
2047:
2043:
2035:
2031:
2008:
1992:
1985:
1981:
1978:
1968:
1952:
1911:
1906:
1894:
1890:
1883:
1878:
1874:
1851:
1847:
1842:
1838:
1831:
1826:
1822:
1820:
1817:
1816:
1810:
1799:
1790:
1786:
1780:
1774:
1770:
1761:
1752:
1749:
1746:
1745:
1743:
1742:
1739:
1738:
1735:
1729:
1726:
1718:
1705:
1702:
1693:
1686:
1679:
1673:
1669:
1662:
1658:
1651:
1647:
1638:
1631:
1623:
1620:
1614:
1611:
1605:
1602:
1596:
1593:neutral element
1586:
1580:
1574:
1572:
1566:
1562:
1559:
1551:
1548:
1542:
1538:
1535:
1529:
1526:
1518:
1515:
1509:
1499:
1490:
1487:
1484:
1483:
1481:
1480:
1477:
1476:
1469:
1466:
1460:
1453:
1450:
1444:
1436:
1432:
1425:
1421:
1417:
1413:
1409:
1406:
1385:
1382:
1376:
1373:
1367:
1363:
1357:quadratic forms
1348:
1343:
1340:
1324:quadratic sieve
1255:
1230:
1226:
1185:
1184:
1180:
1174:
1170:
1168:
1165:
1164:
1155:
1142:
1118:Quadratic sieve
1083:Second Category
1075:
1073:General-purpose
1037:
1026:
977:
975:Special-purpose
972:
945:
939:
933:
927:
921:
915:NP-intermediate
889:
879:
873:
863:
852:
849:
844:
843:
841:
832:
821:
814:
808:
792:
789:
788:
772:
769:
768:
764:
761:
733:
725:
717:
699:
666:
645:
641:
640:
629:
614:
610:
609:
578:
562:
558:
557:
556:
552:
551:
547:
539:
536:
535:
526:
520:
513:sub-exponential
506:
498:
489:
478:
472:
466:
463:polynomial time
455:
453:Time complexity
418:
415:
409:
400:
394:
387:
385:
383:
372:
361:
354:
352:
341:
339:
334:
331:â = 18848997159
325:
323:
321:
317:
313:
309:
305:
299:
293:
279:
265:
250:polynomial time
226:
219:
212:
166:elliptic curves
122:
114:
110:
106:
98:
83:
79:
75:
71:
70:. For example,
40:
39:
34:
31:
24:
17:
12:
11:
5:
4123:
4122:
4111:
4110:
4105:
4100:
4095:
4078:
4077:
4075:
4074:
4069:
4064:
4059:
4054:
4049:
4044:
4039:
4034:
4028:
4026:
4022:
4021:
4019:
4018:
4013:
4008:
4003:
3998:
3993:
3987:
3985:
3978:
3977:
3975:
3974:
3969:
3964:
3954:
3948:
3946:
3939:
3938:
3936:
3935:
3930:
3925:
3920:
3915:
3910:
3905:
3900:
3894:
3892:
3888:
3887:
3885:
3884:
3879:
3874:
3869:
3864:
3859:
3854:
3849:
3844:
3839:
3837:Almost perfect
3834:
3828:
3826:
3822:
3821:
3819:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3783:
3778:
3773:
3768:
3763:
3758:
3752:
3750:
3746:
3745:
3738:
3736:
3734:
3733:
3728:
3723:
3718:
3713:
3708:
3702:
3700:
3696:
3695:
3690:
3689:
3682:
3675:
3667:
3658:
3657:
3655:
3654:
3647:
3644:
3643:
3641:
3640:
3635:
3630:
3625:
3620:
3606:
3601:
3596:
3591:
3585:
3583:
3579:
3578:
3576:
3575:
3570:
3565:
3563:TonelliâShanks
3560:
3555:
3549:
3547:
3541:
3540:
3538:
3537:
3532:
3527:
3522:
3516:
3514:
3508:
3507:
3505:
3504:
3499:
3497:Index calculus
3494:
3492:PohligâHellman
3489:
3484:
3479:
3473:
3471:
3465:
3464:
3462:
3461:
3456:
3451:
3446:
3444:Newton-Raphson
3441:
3436:
3431:
3426:
3420:
3418:
3409:
3408:
3406:
3405:
3400:
3395:
3390:
3385:
3380:
3374:
3372:
3370:Multiplication
3366:
3365:
3363:
3362:
3357:
3355:Trial division
3352:
3347:
3342:
3340:Rational sieve
3337:
3330:
3325:
3320:
3312:
3304:
3299:
3294:
3289:
3284:
3278:
3276:
3270:
3269:
3267:
3266:
3261:
3256:
3251:
3246:
3244:Sieve of Atkin
3240:
3238:
3232:
3231:
3229:
3228:
3223:
3218:
3213:
3206:
3199:
3192:
3185:
3180:
3175:
3170:
3168:Elliptic curve
3165:
3160:
3155:
3149:
3147:
3141:
3140:
3130:
3129:
3122:
3115:
3107:
3098:
3097:
3095:
3094:
3089:
3084:
3078:
3076:
3072:
3071:
3069:
3068:
3063:
3058:
3053:
3043:
3032:
3030:
3026:
3025:
3023:
3022:
3017:
3012:
3006:
3004:
3000:
2999:
2997:
2996:
2991:
2986:
2984:Diffie-Hellman
2981:
2975:
2973:
2969:
2968:
2966:
2965:
2960:
2955:
2950:
2945:
2940:
2935:
2929:
2927:
2923:
2922:
2915:
2914:
2907:
2900:
2892:
2886:
2885:
2879:
2869:
2860:
2850:
2842:
2841:External links
2839:
2838:
2837:
2831:
2820:Addison Wesley
2818:(2 ed.).
2809:
2802:
2784:
2760:
2753:
2741:Carl Pomerance
2731:
2728:
2725:
2724:
2703:(3): 483â516.
2677:
2666:(4): 443â454.
2643:
2602:
2579:(2): 101â127.
2559:
2552:
2528:David Bressoud
2516:
2497:
2473:Wigderson, Avi
2460:
2445:
2382:
2368:
2332:
2310:
2288:Arora, Sanjeev
2279:
2265:
2234:
2203:
2200:on 2019-12-02.
2185:
2173:
2142:
2141:
2139:
2136:
2135:
2134:
2128:
2120:
2115:
2110:
2105:
2099:
2092:
2089:
2082:
2071:
2068:
2059:
2058:
2045:
2028:
1976:
1949:
1948:
1947:
1936:
1933:
1929:
1923:
1920:
1917:
1914:
1909:
1905:
1897:
1893:
1889:
1886:
1882:
1877:
1873:
1869:
1863:
1860:
1857:
1854:
1850:
1841:
1837:
1834:
1830:
1825:
1808:
1795:
1783:
1778:
1767:
1733:
1722:
1715:
1698:
1691:
1684:
1677:
1666:
1643:
1618:
1609:
1600:
1584:
1570:
1555:
1546:
1533:
1522:
1513:
1464:
1448:
1405:
1402:
1380:
1371:
1346:
1339:
1336:
1308:
1307:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1229:
1225:
1221:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1193:
1190:
1183:
1177:
1173:
1154:
1151:
1150:
1149:
1141:
1138:
1137:
1136:
1130:
1125:
1123:Rational sieve
1120:
1115:
1109:
1074:
1071:
1070:
1069:
1064:
1059:
1054:
1049:
1018:
1008:
1003:
1001:Trial division
994:trial division
989:First Category
976:
973:
971:
968:
911:co-NP-complete
796:
776:
755:
696:
695:
684:
680:
673:
670:
664:
660:
657:
654:
651:
648:
644:
636:
633:
627:
623:
620:
617:
613:
607:
603:
600:
597:
594:
591:
585:
582:
576:
570:
567:
561:
555:
550:
546:
543:
454:
451:
408:
405:
290:trial division
211:
208:
103:trial division
35:
32:
26:
15:
9:
6:
4:
3:
2:
4121:
4120:
4109:
4108:Factorization
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4090:
4088:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4029:
4027:
4023:
4017:
4014:
4012:
4011:Polydivisible
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3988:
3986:
3983:
3979:
3973:
3970:
3968:
3965:
3962:
3958:
3955:
3953:
3950:
3949:
3947:
3944:
3940:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3914:
3913:Superabundant
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3895:
3893:
3889:
3883:
3882:ErdĆsâNicolas
3880:
3878:
3875:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3840:
3838:
3835:
3833:
3830:
3829:
3827:
3823:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3791:Perfect power
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3753:
3751:
3747:
3742:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3703:
3701:
3697:
3688:
3683:
3681:
3676:
3674:
3669:
3668:
3665:
3652:
3649:
3648:
3645:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3618:
3614:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3586:
3584:
3580:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3558:Pocklington's
3556:
3554:
3551:
3550:
3548:
3546:
3542:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3517:
3515:
3513:
3509:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3474:
3472:
3470:
3466:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3421:
3419:
3417:
3414:
3410:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3375:
3373:
3371:
3367:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3335:
3331:
3329:
3326:
3324:
3321:
3319:
3317:
3313:
3311:
3309:
3305:
3303:
3302:Pollard's rho
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3279:
3277:
3275:
3271:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3241:
3239:
3237:
3233:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3211:
3207:
3205:
3204:
3200:
3198:
3197:
3193:
3191:
3190:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3150:
3148:
3146:
3142:
3138:
3135:
3128:
3123:
3121:
3116:
3114:
3109:
3108:
3105:
3093:
3090:
3088:
3085:
3083:
3080:
3079:
3077:
3073:
3067:
3064:
3062:
3059:
3057:
3054:
3051:
3047:
3044:
3041:
3037:
3034:
3033:
3031:
3027:
3021:
3018:
3016:
3013:
3011:
3008:
3007:
3005:
3001:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2974:
2970:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2930:
2928:
2924:
2920:
2913:
2908:
2906:
2901:
2899:
2894:
2893:
2890:
2883:
2880:
2878:
2876:
2870:
2868:
2864:
2861:
2859:
2855:
2851:
2848:
2845:
2844:
2834:
2828:
2825:
2821:
2817:
2816:
2810:
2805:
2799:
2795:
2794:
2789:
2785:
2782:
2781:0-201-89684-2
2778:
2774:
2770:
2769:
2764:
2761:
2756:
2754:0-387-94777-9
2750:
2746:
2742:
2738:
2734:
2733:
2720:
2716:
2711:
2706:
2702:
2698:
2691:
2684:
2682:
2673:
2669:
2665:
2661:
2654:
2647:
2639:
2635:
2630:
2625:
2621:
2617:
2613:
2606:
2598:
2594:
2590:
2586:
2582:
2578:
2574:
2570:
2563:
2555:
2549:
2545:
2540:
2539:
2533:
2529:
2523:
2521:
2513:
2508:
2504:
2500:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2464:
2456:
2449:
2441:
2437:
2433:
2429:
2425:
2421:
2417:
2413:
2408:
2403:
2399:
2395:
2394:
2386:
2371:
2365:
2361:
2357:
2353:
2349:
2345:
2344:
2336:
2329:
2325:
2321:
2317:
2313:
2307:
2303:
2299:
2295:
2294:
2289:
2283:
2276:
2272:
2268:
2262:
2258:
2254:
2250:
2249:
2244:
2238:
2230:
2226:
2222:
2215:
2207:
2199:
2195:
2189:
2176:
2170:
2166:
2162:
2158:
2154:
2147:
2143:
2132:
2129:
2127:
2121:
2119:
2116:
2114:
2113:Factorization
2111:
2109:
2106:
2103:
2100:
2098:
2095:
2094:
2088:
2085:
2077:
2067:
2065:
2054:
2041:
2029:
2024:
2020:
2016:
2012:
2004:
2000:
1996:
1989:
1975:
1971:
1964:
1960:
1956:
1950:
1934:
1931:
1927:
1918:
1912:
1907:
1903:
1891:
1887:
1884:
1880:
1875:
1871:
1867:
1858:
1852:
1848:
1839:
1835:
1832:
1828:
1823:
1815:
1814:
1807:
1803:
1798:
1794:
1784:
1777:
1768:
1755:
1732:
1725:
1721:
1716:
1712:
1708:
1701:
1697:
1690:
1683:
1676:
1672:first primes
1667:
1655:
1645:
1644:
1642:
1635:
1629:
1617:
1608:
1599:
1594:
1589:
1583:
1569:
1558:
1554:
1545:
1532:
1525:
1521:
1512:
1507:
1493:
1475:
1463:
1457:
1447:
1442:
1429:
1401:
1397:
1393:
1389:
1379:
1370:
1362:
1358:
1354:
1349:
1335:
1333:
1329:
1325:
1321:
1317:
1313:
1287:
1284:
1281:
1278:
1275:
1266:
1263:
1260:
1246:
1240:
1237:
1234:
1227:
1223:
1219:
1212:
1206:
1203:
1200:
1197:
1191:
1188:
1181:
1175:
1171:
1163:
1162:
1161:
1160:
1147:
1144:
1143:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1113:
1110:
1108:
1105:
1104:
1103:
1101:
1097:
1093:
1090:
1089:
1084:
1080:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1044:
1040:
1033:
1029:
1022:
1019:
1016:
1012:
1009:
1007:
1004:
1002:
999:
998:
997:
995:
990:
986:
981:
967:
965:
961:
957:
953:
948:
944:of digits of
942:
936:
930:
924:
918:
916:
912:
907:
905:
901:
897:
892:
887:
882:
876:
870:
866:
855:
847:
839:
835:
830:
826:
820:
817:
811:
794:
774:
754:
752:
748:
743:
741:
736:
729:
721:
715:
711:
707:
702:
682:
678:
671:
668:
662:
658:
655:
652:
649:
646:
642:
634:
631:
625:
621:
618:
615:
611:
605:
598:
592:
589:
583:
580:
574:
568:
565:
559:
553:
548:
544:
541:
534:
533:
532:
529:
523:
518:
514:
509:
502:
495:
492:
485:
481:
475:
469:
464:
460:
450:
448:
444:
440:
435:
433:
428:
426:
421:
414:
404:
397:
380:= 18848997161
379:
375:
369:= 18848997157
368:
364:
348:
344:
337:
328:
302:
296:
291:
286:
282:
276:
272:
268:
262:
261:prime factors
257:
255:
251:
247:
243:
242:empty product
239:
235:
222:
216:
207:
205:
201:
197:
192:
190:
186:
182:
177:
175:
171:
167:
163:
159:
155:
151:
147:
143:
139:
135:
130:
128:
120:
104:
95:
93:
89:
69:
65:
61:
57:
53:
49:
45:
38:
22:
4072:Superperfect
4067:Refactorable
3862:Superperfect
3857:Hyperperfect
3842:Quasiperfect
3726:Prime factor
3705:
3650:
3332:
3315:
3307:
3273:
3226:MillerâRabin
3208:
3201:
3194:
3189:LucasâLehmer
3187:
2932:
2874:
2853:
2813:
2792:
2772:
2771:, Volume 2:
2766:
2763:Donald Knuth
2747:. Springer.
2744:
2700:
2696:
2663:
2659:
2646:
2619:
2615:
2605:
2597:the original
2576:
2572:
2562:
2537:
2488:
2485:Leader, Imre
2463:
2448:
2397:
2391:
2385:
2373:. Retrieved
2342:
2335:
2292:
2282:
2247:
2237:
2220:
2206:
2198:the original
2188:
2178:, retrieved
2156:
2146:
2080:
2073:
2061:
2052:
2022:
2018:
2014:
2010:
2002:
1998:
1994:
1987:
1973:
1969:
1962:
1958:
1954:
1813:satisfying:
1805:
1801:
1796:
1792:
1775:
1753:
1730:
1723:
1719:
1710:
1706:
1699:
1695:
1688:
1681:
1674:
1653:
1636:
1615:
1606:
1597:
1590:
1581:
1567:
1556:
1552:
1543:
1530:
1523:
1519:
1510:
1491:
1461:
1458:
1445:
1427:
1407:
1395:
1391:
1387:
1377:
1368:
1361:discriminant
1344:
1341:
1327:
1309:
1159:running time
1156:
1091:
1087:
1082:
1078:
1076:
1038:
1027:
988:
984:
982:
978:
946:
940:
934:
928:
922:
919:
908:
890:
880:
874:
868:
864:
853:
845:
837:
833:
822:
815:
809:
756:
744:
734:
727:
719:
700:
697:
527:
525:-bit number
521:
507:
500:
496:
490:
483:
473:
471:-bit number
467:
456:
436:
429:
419:
416:
395:
377:
373:
366:
362:
346:
342:
335:
326:
300:
294:
284:
280:
274:
270:
266:
258:
231:
220:
193:
178:
146:cryptography
131:
96:
87:
68:prime number
47:
41:
3996:Extravagant
3991:Equidigital
3952:Untouchable
3872:Semiperfect
3852:Hemiperfect
3781:Square-free
3482:Pollard rho
3439:Goldschmidt
3173:Pocklington
3163:BaillieâPSW
2943:RSA problem
2512:p. 583
1704:, for some
1366:denoted by
1353:class group
1096:RSA numbers
950:) with the
819:besides 1?
716:takes only
708:, however,
511:, that is,
388:18848997157
314:18848997161
196:RSA problem
158:mathematics
134:non-quantum
119:square root
44:mathematics
4087:Categories
4032:Arithmetic
4025:Other sets
3984:-dependent
3594:Cornacchia
3589:Chakravala
3137:algorithms
2948:Strong RSA
2938:Phi-hiding
2730:References
2532:Stan Wagon
2180:2022-06-22
1694:= 5, ...,
1573:(log|
1506:generators
1459:Denote by
1316:L-notation
1079:Category 2
1036:Williams'
1025:Pollard's
985:Category 1
710:Peter Shor
425:semiprimes
417:Among the
411:See also:
391:â = 137292
204:public-key
181:semiprimes
142:difficulty
4062:Descartes
4037:Deficient
3972:Betrothed
3877:Practical
3766:Semiprime
3761:Composite
3568:Berlekamp
3525:Euclidean
3413:Euclidean
3393:ToomâCook
3388:Karatsuba
2360:1887/2149
2328:215746906
1984:in which
1896:Δ
1888:∈
1881:∏
1836:∈
1829:∏
1668:Take the
1443:forms in
1285:
1279:
1264:
1088:Kraitchik
1043:algorithm
1032:algorithm
732:space on
724:time and
656:
650:
619:
545:
531:in time:
459:algorithm
443:Xeon Gold
138:algorithm
4047:Solitary
4042:Friendly
3967:Sociable
3957:Amicable
3945:-related
3898:Abundant
3796:Achilles
3786:Powerful
3699:Overview
3535:Lehmer's
3429:Chunking
3416:division
3345:Fermat's
3029:Lattices
3003:Pairings
2858:download
2790:(2013).
2743:(2001).
2534:(2000).
2487:(eds.),
2432:11780055
2375:12 March
2245:(2011),
2091:See also
1800: :
1657:, where
1431:, where
1322:and the
1312:little-o
1135:(SQUFOF)
1102:method.
477:in time
316:, where
269:= 171 Ă
152:and the
148:such as
4052:Sublime
4006:Harshad
3832:Perfect
3816:Unusual
3806:Regular
3776:Sphenic
3711:Divisor
3651:Italics
3573:Kunerth
3553:Cipolla
3434:Fourier
3403:FĂŒrer's
3297:Euler's
3287:Dixon's
3210:PĂ©pin's
2719:1137100
2638:0878705
2593:0657269
2507:2467561
2440:4400832
2412:Bibcode
2320:2500087
2275:2789493
2040:2-Sylow
1758:
1744:
1577:|)
1496:
1482:
1114:(CFRAC)
858:
842:
807:, does
439:RSA-250
432:RSA-240
401:1372933
386:√
353:√
340:√
324:√
310:1372933
246:Testing
232:By the
202:-based
60:factors
56:product
54:into a
4001:Frugal
3961:Triple
3801:Smooth
3771:Pronic
3633:Schoof
3520:Binary
3424:Binary
3360:Shor's
3178:Fermat
2847:msieve
2829:
2800:
2779:
2751:
2717:
2636:
2591:
2550:
2544:168â69
2505:
2495:
2438:
2430:
2393:Nature
2366:
2326:
2318:
2308:
2273:
2263:
2171:
2042:group
1986:Î = â4
1441:smooth
1092:family
1045:, and
894:. The
760:
312:, and
278:where
172:, and
78:, but
4016:Smith
3933:Weird
3811:Rough
3756:Prime
3454:Short
3183:Lucas
2693:(PDF)
2656:(PDF)
2436:S2CID
2402:arXiv
2324:S2CID
2217:(PDF)
2138:Notes
2009:Î = (
1737:with
1687:= 3,
1680:= 2,
1652:Î = â
1537:with
1472:with
1426:Î = â
1085:, or
862:with
829:co-NP
306:13729
283:<
227:2 Ă 3
223:= 864
3982:Base
3449:Long
3383:Long
2827:ISBN
2798:ISBN
2777:ISBN
2749:ISBN
2739:and
2548:ISBN
2530:and
2493:ISBN
2428:PMID
2377:2021
2364:ISBN
2306:ISBN
2261:ISBN
2169:ISBN
1993:Î =
1789:and
1717:Let
1646:Let
1637:Let
1314:and
827:and
787:and
393:for
371:and
185:bits
160:and
3613:LLL
3459:SRT
3318:+ 1
3310:â 1
3158:APR
3153:AKS
3050:gap
3040:gap
2705:doi
2668:doi
2624:doi
2581:doi
2420:doi
2398:414
2356:hdl
2348:doi
2298:doi
2253:doi
2225:doi
2161:doi
2055:(Î)
2050:of
2048:(Î)
2044:Sll
2021:+ 2
2013:â 2
2007:or
2001:â 4
1991:or
1773:of
1764:= 1
1628:gcd
1595:of
1541:in
1528:of
1508:of
1502:= 1
1359:of
1310:in
1282:log
1276:log
1261:log
1041:+ 1
1030:â 1
987:or
964:RSA
904:BQP
740:NMR
653:log
647:log
616:log
542:exp
457:No
358:= 2
244:.)
225:as
200:RSA
121:of
42:In
4089::
3617:KZ
3615:;
2822:-
2765:.
2715:MR
2713:.
2699:.
2695:.
2680:^
2664:50
2662:.
2658:.
2634:MR
2632:.
2620:48
2618:.
2614:.
2589:MR
2587:.
2575:.
2571:.
2546:.
2519:^
2503:MR
2501:,
2483:;
2479:;
2471:;
2434:.
2426:.
2418:.
2410:.
2396:.
2362:.
2354:.
2322:,
2316:MR
2314:,
2304:,
2271:MR
2269:,
2259:,
2167:,
2087:.
2066:.
2017:)(
1988:ac
1972:â
1961:,
1957:,
1935:1.
1804:â
1709:â
1654:dn
1588:.
1428:dn
1424:,
1394:,
1390:,
1375:.
1334:.
1081:,
1034:,
900:UP
867:â€
836:=
825:NP
753:.
726:O(
718:O(
503:))
403:.
376:+
365:â
351:=
345:â
338:=
308:,
273:Ă
176:.
168:,
113:,
109:,
94:.
72:15
46:,
3963:)
3959:(
3686:e
3679:t
3672:v
3619:)
3611:(
3316:p
3308:p
3126:e
3119:t
3112:v
3052:)
3048:(
3042:)
3038:(
2911:e
2904:t
2897:v
2835:.
2808:.
2806:.
2757:.
2721:.
2707::
2701:5
2674:.
2670::
2640:.
2626::
2583::
2577:3
2556:.
2514:.
2457:.
2442:.
2422::
2414::
2404::
2379:.
2358::
2350::
2300::
2255::
2231:.
2227::
2163::
2124:p
2083:n
2081:L
2057:.
2053:G
2046:2
2036:n
2032:n
2027:.
2025:)
2023:a
2019:b
2015:a
2011:b
2005:)
2003:c
1999:a
1997:(
1995:a
1982:Î
1977:Î
1974:G
1970:f
1965:)
1963:c
1959:b
1955:a
1953:(
1932:=
1928:)
1922:)
1919:q
1916:(
1913:t
1908:q
1904:f
1892:P
1885:q
1876:(
1872:.
1868:)
1862:)
1859:x
1856:(
1853:r
1849:x
1840:X
1833:x
1824:(
1811:}
1809:Î
1806:P
1802:q
1797:q
1793:f
1791:{
1787:X
1782:.
1779:Î
1776:G
1771:X
1766:.
1762:)
1754:q
1750:/
1747:Î
1740:(
1734:Î
1731:G
1724:q
1720:f
1714:.
1711:N
1707:t
1700:t
1696:p
1692:3
1689:p
1685:2
1682:p
1678:1
1675:p
1670:t
1663:Î
1659:d
1648:Î
1639:n
1632:n
1624:Î
1619:Î
1616:G
1610:Î
1607:G
1601:Î
1598:G
1585:0
1582:c
1575:Î
1571:0
1568:c
1563:q
1557:q
1553:f
1547:Î
1544:P
1539:q
1534:Î
1531:G
1524:q
1520:f
1514:Î
1511:G
1500:)
1492:q
1488:/
1485:Î
1478:(
1470:q
1465:Î
1462:P
1454:d
1449:Î
1446:G
1437:d
1433:d
1422:n
1418:Î
1414:n
1410:n
1398:)
1396:c
1392:b
1388:a
1386:(
1381:Î
1378:G
1372:Î
1369:G
1364:Î
1347:n
1345:L
1291:)
1288:n
1273:(
1270:)
1267:n
1258:(
1253:)
1250:)
1247:1
1244:(
1241:o
1238:+
1235:1
1232:(
1228:e
1224:=
1220:]
1216:)
1213:1
1210:(
1207:o
1204:+
1201:1
1198:,
1192:2
1189:1
1182:[
1176:n
1172:L
1039:p
1028:p
947:n
941:b
935:n
929:n
923:n
891:n
881:k
875:n
869:k
865:d
860:)
854:d
850:/
846:n
840:(
838:d
834:n
816:k
810:n
795:k
775:n
735:b
730:)
728:b
722:)
720:b
701:n
683:.
679:)
672:3
669:2
663:)
659:n
643:(
635:3
632:1
626:)
622:n
612:(
606:)
602:)
599:1
596:(
593:o
590:+
584:3
581:2
575:)
569:3
566:8
560:(
554:(
549:(
528:n
522:b
508:Δ
501:Δ
491:k
486:)
484:b
482:(
480:O
474:n
468:b
420:b
396:a
384:â
378:b
374:a
367:b
363:a
355:4
347:n
343:a
336:b
327:n
322:â
301:n
295:p
285:q
281:p
275:q
271:p
267:n
221:n
123:n
115:5
111:3
107:2
99:n
80:7
30::
23:.
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