33:
1127:
386:
3408:
1653:
414:
In its simplest form, Fermat's method might be even slower than trial division (worst case). Nonetheless, the combination of trial division and Fermat's is more effective than either by itself.
1523:
3412:
1735:
1694:
1247:
588:
2920:
2660:, which are the "worst-case". The primary improvement that quadratic sieve makes over Fermat's factorization method is that instead of simply finding a square in the sequence of
1582:
2405:
197:
470:
2543:
2499:
2092:
1832:
1547:
1472:
1328:
1271:
874:
2638:
910:
251:
50:
996:
2953:
2691:
2001:
1956:
781:
749:
524:
2220:
2148:
2059:
2028:
1859:
1189:
946:
713:
687:
287:
2590:
2361:
2330:
1160:
818:
2455:
2432:
1001:
299:
1477:
Trial division would normally try up to 48,432; but after only four Fermat steps, we need only divide up to 47830, to find a factor or prove primality.
3468:
3277:
2913:
2769:
3145:
97:
69:
3087:
2906:
1587:
3016:
3193:
2795:
76:
2991:
116:
2003:
produces 3, 4, 7, 8, 12, and 19 modulo 20 for these values. It is apparent that only the 4 from this list can be a square. Thus,
3102:
3077:
83:
3021:
2984:
3282:
3173:
3092:
3082:
2754:
2958:
3110:
54:
3363:
65:
3358:
3287:
3188:
3325:
2719:
2278:
3239:
1483:
2774:
2886:
2697:
is a square, and it does this in a highly efficient manner. The end result is the same: a difference of squares mod
3404:
3394:
3353:
3129:
3123:
3097:
2968:
2803:
2653:
3389:
3330:
786:
Suppose N has more than two prime factors. That procedure first finds the factorization with the least values of
144:
3292:
3165:
3011:
2963:
1699:
1658:
1202:
90:
3307:
3198:
43:
541:
3418:
3368:
3348:
2892:
2267:
b2 is a square, modulo modulus: FermatSieve(N, a, aend, astep * modulus, NextModulus)
3069:
3044:
2973:
2759:
2714:
1552:
3428:
2854:
2596:. R. Lehman devised a systematic way to do this, so that Fermat's plus trial division can factor N in
3423:
3315:
3297:
3272:
3234:
2978:
2739:
2734:
403:
are also odd, so those halves are integers. (A multiple of four is also a difference of squares: let
137:
2366:
1740:
In this regard, Fermat's method gives diminishing returns. One would surely stop before this point:
153:
3433:
3399:
3320:
3224:
3183:
3178:
3155:
3059:
429:
2504:
2460:
2071:
1811:
487:
a square: a â a + 1 b2 â a*a - N // equivalently: // b2 â b2 + 2*a + 1
3264:
3211:
3208:
3049:
2948:
2822:
1963:
1528:
1445:
1309:
1252:
827:
3005:
2998:
2599:
883:
209:
955:
3384:
3340:
3054:
3031:
2764:
2663:
1973:
1928:
754:
3229:
2866:
2812:
722:
503:
133:
2198:
2126:
2037:
2006:
1837:
3219:
3118:
2649:
2333:
1165:
1122:{\displaystyle (c+d)/2-{\sqrt {N}}=({\sqrt {d}}-{\sqrt {c}})^{2}/2=({\sqrt {N}}-c)^{2}/2c}
922:
692:
666:
263:
2567:
2338:
2307:
1139:
797:
2437:
2414:
3249:
3150:
3135:
3039:
2940:
2724:
3462:
3244:
2929:
2749:
381:{\displaystyle N=\left({\frac {c+d}{2}}\right)^{2}-\left({\frac {c-d}{2}}\right)^{2}}
1195:
has a factor close to its square root, the method works quickly. More precisely, if
3254:
2744:
2871:
1970:
by 10. In this example, N is 17 mod 20, so subtracting 17 mod 20 (or adding 3),
32:
2898:
2657:
590:. Since 125 is not a square, a second try is made by increasing the value of
17:
2932:
2729:
2648:
The fundamental ideas of Fermat's factorization method are the basis of the
2297:
Fermat's method works best when there is a factor near the square-root of
2285:-values remain; that is, when (aend-astart)/astep is small. Also, because
2289:
s step-size is constant, one can compute successive b2's with additions.
594:
by 1. The second attempt also fails, because 282 is again not a square.
1273:, the method requires only one step; this is independent of the size of
2826:
203:
140:
2245:
One generally chooses a power of a different prime for each modulus.
1696:
the bound for trial division is 47830. A reasonable choice could be
1648:{\displaystyle a_{\mathrm {max} }-{\sqrt {a_{\mathrm {max} }^{2}-N}}}
2817:
2252:-values (start, end, and step) and a modulus, one can proceed thus:
1330:
rounded up to the next integer, which is 48,433, we can tabulate:
1480:
This all suggests a combined factoring method. Choose some bound
2255:
FermatSieve(N, astart, aend, astep, modulus) a â astart
253:; if neither factor equals one, it is a proper factorization of
2902:
26:
1434:
In practice, one wouldn't bother with that last row until
663:
The third try produces the perfect square of 441. Thus,
3449:
indicate that algorithm is for numbers of special forms
2693:, it finds a subset of elements of this sequence whose
2068:
This can be performed with any modulus. Using the same
1925:
It is not necessary to compute all the square-roots of
2592:
values can be tried, and try to factor each resulting
260:
Each odd number has such a representation. Indeed, if
2666:
2602:
2570:
2507:
2463:
2440:
2417:
2369:
2341:
2310:
2201:
2129:
2074:
2040:
2009:
1976:
1931:
1840:
1814:
1702:
1661:
1590:
1555:
1531:
1486:
1448:
1312:
1255:
1205:
1168:
1142:
1004:
958:
925:
886:
830:
800:
757:
725:
695:
669:
544:
506:
432:
302:
266:
212:
156:
1962:. Squares are always congruent to 0, 1, 4, 5, 9, 16
1191:
steps. This is a bad way to prove primality. But if
3377:
3339:
3306:
3263:
3207:
3164:
3068:
3030:
2939:
57:. Unsourced material may be challenged and removed.
2685:
2632:
2584:
2537:
2493:
2449:
2426:
2399:
2355:
2324:
2214:
2142:
2086:
2053:
2022:
1995:
1950:
1853:
1834:, one can quickly tell that none of the values of
1826:
1729:
1688:
1647:
1584:. This gives a bound for trial division which is
1576:
1541:
1517:
1466:
1322:
1265:
1241:
1183:
1154:
1121:
990:
940:
904:
868:
812:
775:
743:
707:
681:
582:
518:
464:
380:
281:
245:
191:
1518:{\displaystyle a_{\mathrm {max} }>{\sqrt {N}}}
2656:, the best-known algorithms for factoring large
2508:
2464:
1474:, Fermat's method would have found it already.
2564:Generally, if the ratio is not known, various
2914:
8:
2034:is 1, 9, 11 or 19 mod 20; it will produce a
1966:20. The values repeat with each increase of
820:is the smallest factor â„ the square-root of
2701:that, if nontrivial, can be used to factor
1285:Consider trying to factor the prime number
2921:
2907:
2899:
1525:; use Fermat's method for factors between
998:, so the number of steps is approximately
479:a â ceiling(sqrt(N)) b2 â a*a - N
2870:
2816:
2671:
2665:
2617:
2613:
2601:
2574:
2569:
2506:
2462:
2439:
2416:
2368:
2345:
2340:
2314:
2309:
2304:If the approximate ratio of two factors (
2206:
2200:
2134:
2128:
2073:
2045:
2039:
2014:
2008:
1981:
1975:
1936:
1930:
1845:
1839:
1813:
1708:
1707:
1701:
1667:
1666:
1660:
1631:
1619:
1618:
1612:
1596:
1595:
1589:
1561:
1560:
1554:
1532:
1530:
1508:
1492:
1491:
1485:
1447:
1313:
1311:
1256:
1254:
1229:
1225:
1207:
1204:
1167:
1141:
1108:
1102:
1085:
1071:
1065:
1054:
1044:
1031:
1020:
1003:
980:
957:
924:
885:
846:
829:
799:
756:
724:
694:
668:
562:
549:
543:
534:rounded up to the next integer, which is
505:
456:
437:
431:
372:
350:
336:
314:
301:
265:
211:
180:
167:
155:
117:Learn how and when to remove this message
2893:Fermat's Factorization Online Calculator
2770:Table of Gaussian integer factorizations
2061:which ends in 4 mod 20 and, if square,
1863:
1742:
1730:{\displaystyle a_{\mathrm {max} }=55000}
1689:{\displaystyle a_{\mathrm {max} }=48436}
1332:
596:
2786:
1242:{\displaystyle {\left(4N\right)}^{1/4}}
136:, is based on the representation of an
1958:, nor even examine all the values for
583:{\displaystyle b^{2}=78^{2}-5959=125}
7:
2855:"Speeding Fermat's factoring method"
55:adding citations to reliable sources
2887:Fermat's factorization running time
2030:must be 1 mod 20, which means that
1438:is an integer. But observe that if
2407:, and Fermat's method, applied to
1715:
1712:
1709:
1674:
1671:
1668:
1626:
1623:
1620:
1603:
1600:
1597:
1577:{\displaystyle a_{\mathrm {max} }}
1568:
1565:
1562:
1499:
1496:
1493:
1280:
25:
3130:Special number field sieve (SNFS)
3124:General number field sieve (GNFS)
3469:Integer factorization algorithms
31:
2363:can be picked near that value.
2263:: b2 â a*a - N
1808:When considering the table for
952:be the largest subroot factor.
66:"Fermat's factorization method"
42:needs additional citations for
2627:
2606:
2526:
2511:
2482:
2467:
2400:{\displaystyle Nuv=cv\cdot du}
1655:. In the above example, with
1178:
1172:
1099:
1082:
1062:
1041:
1017:
1005:
977:
965:
863:
851:
240:
228:
225:
213:
192:{\displaystyle N=a^{2}-b^{2}.}
1:
2872:10.1090/S0025-5718-99-01133-3
2167:3 or 5 or 11 or 13 modulo 16
876:is the largest factor †root-
465:{\displaystyle a^{2}-N=b^{2}}
130:Fermat's factorization method
3088:Lenstra elliptic curve (ECM)
2755:Euler's factorization method
2720:Factorization of polynomials
2538:{\displaystyle \gcd(N,du)=d}
2494:{\displaystyle \gcd(N,cv)=c}
2087:{\displaystyle N=2345678917}
1827:{\displaystyle N=2345678917}
422:One tries various values of
2794:Lehman, R. Sherman (1974).
2065:will end in 2 or 8 mod 10.
1542:{\displaystyle {\sqrt {N}}}
1467:{\displaystyle a-b=47830.1}
1442:had a subroot factor above
1323:{\displaystyle {\sqrt {N}}}
1306:throughout. Going up from
1281:Fermat's and trial division
1266:{\displaystyle {\sqrt {N}}}
869:{\displaystyle a-b=N/(a+b)}
3485:
3395:Exponentiation by squaring
3078:Continued fraction (CFRAC)
2859:Mathematics of Computation
2848:, vol. 2, p. 256
2804:Mathematics of Computation
2796:"Factoring Large Integers"
2654:general number field sieve
2633:{\displaystyle O(N^{1/3})}
905:{\displaystyle N=1\cdot N}
246:{\displaystyle (a+b)(a-b)}
3442:
1737:giving a bound of 28937.
991:{\displaystyle a=(c+d)/2}
880:. If the procedure finds
145:difference of two squares
2411:, will find the factors
3308:Greatest common divisor
2686:{\displaystyle a^{2}-n}
1996:{\displaystyle a^{2}-N}
1951:{\displaystyle a^{2}-N}
776:{\displaystyle a+b=101}
500:For example, to factor
3419:Modular exponentiation
2687:
2634:
2586:
2539:
2495:
2451:
2428:
2401:
2357:
2326:
2293:Multiplier improvement
2216:
2144:
2088:
2055:
2024:
1997:
1952:
1855:
1828:
1731:
1690:
1649:
1578:
1543:
1519:
1468:
1324:
1267:
1243:
1185:
1156:
1123:
992:
942:
906:
870:
814:
777:
745:
744:{\displaystyle a-b=59}
709:
683:
584:
530:is the square root of
520:
519:{\displaystyle N=5959}
466:
382:
289:is a factorization of
283:
247:
193:
3146:Shanks's square forms
3070:Integer factorization
3045:Sieve of Eratosthenes
2760:Integer factorization
2715:Completing the square
2688:
2635:
2587:
2540:
2496:
2452:
2429:
2402:
2358:
2327:
2217:
2215:{\displaystyle a^{2}}
2145:
2143:{\displaystyle a^{2}}
2089:
2056:
2054:{\displaystyle b^{2}}
2025:
2023:{\displaystyle a^{2}}
1998:
1953:
1856:
1854:{\displaystyle b^{2}}
1829:
1732:
1691:
1650:
1579:
1544:
1520:
1469:
1325:
1268:
1244:
1186:
1157:
1124:
993:
943:
907:
871:
815:
778:
746:
715:, and the factors of
710:
684:
585:
521:
467:
383:
284:
248:
194:
3424:Montgomery reduction
3298:Function field sieve
3273:Baby-step giant-step
3119:Quadratic sieve (QS)
2775:Unique factorization
2735:Monoid factorisation
2664:
2600:
2568:
2505:
2461:
2438:
2415:
2367:
2339:
2308:
2281:is stopped when few
2248:Given a sequence of
2199:
2127:
2072:
2038:
2007:
1974:
1929:
1838:
1812:
1700:
1659:
1588:
1553:
1529:
1484:
1446:
1310:
1253:
1203:
1184:{\displaystyle O(N)}
1166:
1140:
1002:
956:
941:{\displaystyle N=cd}
923:
884:
828:
798:
755:
723:
708:{\displaystyle b=21}
693:
682:{\displaystyle a=80}
667:
542:
526:, the first try for
504:
430:
300:
282:{\displaystyle N=cd}
264:
210:
154:
51:improve this article
3434:Trachtenberg system
3400:Integer square root
3341:Modular square root
3060:Wheel factorization
3012:Quadratic Frobenius
2992:LucasâLehmerâRiesel
2895:, at windowspros.ru
2585:{\displaystyle u/v}
2356:{\displaystyle v/u}
2332:) is known, then a
2325:{\displaystyle d/c}
1636:
1292:, but also compute
1155:{\displaystyle c=1}
813:{\displaystyle a+b}
202:That difference is
3326:Extended Euclidean
3265:Discrete logarithm
3194:SchönhageâStrassen
3050:Sieve of Pritchard
2865:(228): 1729â1737.
2683:
2644:Other improvements
2630:
2582:
2535:
2491:
2450:{\displaystyle du}
2447:
2427:{\displaystyle cv}
2424:
2397:
2353:
2322:
2271:a â a + astep
2212:
2140:
2084:
2051:
2020:
1993:
1948:
1851:
1824:
1727:
1686:
1645:
1614:
1574:
1539:
1515:
1464:
1320:
1263:
1239:
1199:differs less than
1181:
1152:
1136:is prime (so that
1119:
988:
938:
912:, that shows that
902:
866:
810:
773:
741:
705:
679:
580:
516:
496:// or a + sqrt(b2)
477:// N should be odd
462:
378:
279:
243:
189:
3456:
3455:
3055:Sieve of Sundaram
2853:McKee, J (1999).
2846:Oeuvres de Fermat
2765:Program synthesis
2740:Pascal's triangle
2243:
2242:
1923:
1922:
1804:Sieve improvement
1801:
1800:
1643:
1537:
1513:
1432:
1431:
1318:
1261:
1090:
1059:
1049:
1036:
661:
660:
490:// a â a + 1
475:FermatFactor(N):
366:
330:
127:
126:
119:
101:
16:(Redirected from
3476:
3405:Integer relation
3378:Other algorithms
3283:Pollard kangaroo
3174:Ancient Egyptian
3032:Prime-generating
3017:SolovayâStrassen
2930:Number-theoretic
2923:
2916:
2909:
2900:
2889:, at blogspot.in
2876:
2874:
2849:
2831:
2830:
2820:
2811:(126): 637â646.
2800:
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2536:
2500:
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2448:
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2430:
2425:
2406:
2404:
2403:
2398:
2362:
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2359:
2354:
2349:
2331:
2329:
2328:
2323:
2318:
2239:4 or 5 modulo 9
2235:
2221:
2219:
2218:
2213:
2211:
2210:
2163:
2149:
2147:
2146:
2141:
2139:
2138:
2097:
2096:
2093:
2091:
2090:
2085:
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2050:
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2029:
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2026:
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2019:
2018:
2002:
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1999:
1994:
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1985:
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1961:
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1941:
1940:
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1852:
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1583:
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1504:
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1471:
1470:
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1305:
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1224:
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1190:
1188:
1187:
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1158:
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1128:
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1125:
1120:
1112:
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1091:
1086:
1075:
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1069:
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1055:
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1045:
1037:
1032:
1024:
997:
995:
994:
989:
984:
947:
945:
944:
939:
911:
909:
908:
903:
875:
873:
872:
867:
850:
819:
817:
816:
811:
782:
780:
779:
774:
750:
748:
747:
742:
718:
714:
712:
711:
706:
688:
686:
685:
680:
597:
589:
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586:
581:
567:
566:
554:
553:
537:
533:
525:
523:
522:
517:
471:
469:
468:
463:
461:
460:
442:
441:
387:
385:
384:
379:
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371:
367:
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351:
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340:
335:
331:
326:
315:
288:
286:
285:
280:
252:
250:
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244:
198:
196:
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190:
185:
184:
172:
171:
134:Pierre de Fermat
122:
115:
111:
108:
102:
100:
59:
35:
27:
21:
3484:
3483:
3479:
3478:
3477:
3475:
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3473:
3459:
3458:
3457:
3452:
3438:
3373:
3335:
3302:
3259:
3203:
3160:
3064:
3026:
2999:Proth's theorem
2941:Primality tests
2935:
2927:
2883:
2852:
2844:Fermat (1894),
2843:
2840:
2835:
2834:
2818:10.2307/2005940
2798:
2793:
2792:
2788:
2783:
2711:
2667:
2662:
2661:
2650:quadratic sieve
2646:
2609:
2598:
2597:
2566:
2565:
2503:
2502:
2459:
2458:
2436:
2435:
2413:
2412:
2365:
2364:
2337:
2336:
2334:rational number
2306:
2305:
2295:
2275:
2233:
2202:
2197:
2196:
2161:
2130:
2125:
2124:
2070:
2069:
2062:
2041:
2036:
2035:
2031:
2010:
2005:
2004:
1977:
1972:
1971:
1967:
1959:
1932:
1927:
1926:
1841:
1836:
1835:
1810:
1809:
1806:
1703:
1698:
1697:
1662:
1657:
1656:
1591:
1586:
1585:
1556:
1551:
1550:
1527:
1526:
1487:
1482:
1481:
1444:
1443:
1308:
1307:
1297:
1290:= 2,345,678,917
1286:
1283:
1251:
1250:
1212:
1208:
1206:
1201:
1200:
1164:
1163:
1138:
1137:
1098:
1061:
1000:
999:
954:
953:
921:
920:
882:
881:
826:
825:
796:
795:
753:
752:
721:
720:
716:
691:
690:
665:
664:
558:
545:
540:
539:
535:
531:
502:
501:
498:
452:
433:
428:
427:
420:
352:
346:
345:
316:
310:
309:
298:
297:
262:
261:
208:
207:
176:
163:
152:
151:
123:
112:
106:
103:
60:
58:
48:
36:
23:
22:
15:
12:
11:
5:
3482:
3480:
3472:
3471:
3461:
3460:
3454:
3453:
3451:
3450:
3443:
3440:
3439:
3437:
3436:
3431:
3426:
3421:
3416:
3402:
3397:
3392:
3387:
3381:
3379:
3375:
3374:
3372:
3371:
3366:
3361:
3359:TonelliâShanks
3356:
3351:
3345:
3343:
3337:
3336:
3334:
3333:
3328:
3323:
3318:
3312:
3310:
3304:
3303:
3301:
3300:
3295:
3293:Index calculus
3290:
3288:PohligâHellman
3285:
3280:
3275:
3269:
3267:
3261:
3260:
3258:
3257:
3252:
3247:
3242:
3240:Newton-Raphson
3237:
3232:
3227:
3222:
3216:
3214:
3205:
3204:
3202:
3201:
3196:
3191:
3186:
3181:
3176:
3170:
3168:
3166:Multiplication
3162:
3161:
3159:
3158:
3153:
3151:Trial division
3148:
3143:
3138:
3136:Rational sieve
3133:
3126:
3121:
3116:
3108:
3100:
3095:
3090:
3085:
3080:
3074:
3072:
3066:
3065:
3063:
3062:
3057:
3052:
3047:
3042:
3040:Sieve of Atkin
3036:
3034:
3028:
3027:
3025:
3024:
3019:
3014:
3009:
3002:
2995:
2988:
2981:
2976:
2971:
2966:
2964:Elliptic curve
2961:
2956:
2951:
2945:
2943:
2937:
2936:
2928:
2926:
2925:
2918:
2911:
2903:
2897:
2896:
2890:
2882:
2881:External links
2879:
2878:
2877:
2850:
2839:
2836:
2833:
2832:
2785:
2784:
2782:
2779:
2778:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2727:
2725:Factor theorem
2722:
2717:
2710:
2707:
2682:
2679:
2674:
2670:
2645:
2642:
2629:
2624:
2620:
2616:
2612:
2608:
2605:
2581:
2577:
2573:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2457:quickly. Then
2446:
2443:
2423:
2420:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2352:
2348:
2344:
2321:
2317:
2313:
2294:
2291:
2254:
2241:
2240:
2237:
2230:
2227:
2226:
2223:
2209:
2205:
2193:
2190:
2189:
2186:
2183:
2180:
2179:
2178:0, 1, 4, or 7
2176:
2173:
2169:
2168:
2165:
2158:
2155:
2154:
2151:
2137:
2133:
2121:
2118:
2117:
2114:
2111:
2108:
2107:
2106:0, 1, 4, or 9
2104:
2101:
2083:
2080:
2077:
2048:
2044:
2017:
2013:
1992:
1989:
1984:
1980:
1947:
1944:
1939:
1935:
1921:
1920:
1917:
1914:
1911:
1908:
1902:
1901:
1898:
1895:
1892:
1889:
1883:
1882:
1879:
1876:
1873:
1870:
1848:
1844:
1823:
1820:
1817:
1805:
1802:
1799:
1798:
1795:
1792:
1782:
1781:
1778:
1775:
1769:
1768:
1767:1,254,561,087
1765:
1762:
1756:
1755:
1752:
1749:
1726:
1723:
1717:
1714:
1711:
1706:
1685:
1682:
1676:
1673:
1670:
1665:
1642:
1639:
1634:
1628:
1625:
1622:
1617:
1611:
1605:
1602:
1599:
1594:
1570:
1567:
1564:
1559:
1536:
1512:
1507:
1501:
1498:
1495:
1490:
1463:
1460:
1457:
1454:
1451:
1430:
1429:
1426:
1423:
1420:
1417:
1407:
1406:
1403:
1400:
1397:
1394:
1388:
1387:
1384:
1381:
1378:
1375:
1369:
1368:
1365:
1362:
1359:
1356:
1350:
1349:
1346:
1343:
1340:
1337:
1317:
1282:
1279:
1260:
1236:
1232:
1228:
1222:
1218:
1215:
1211:
1180:
1177:
1174:
1171:
1151:
1148:
1145:
1118:
1115:
1111:
1105:
1101:
1097:
1094:
1089:
1084:
1081:
1078:
1074:
1068:
1064:
1058:
1053:
1048:
1043:
1040:
1035:
1030:
1027:
1023:
1019:
1016:
1013:
1010:
1007:
987:
983:
979:
976:
973:
970:
967:
964:
961:
937:
934:
931:
928:
901:
898:
895:
892:
889:
865:
862:
859:
856:
853:
849:
845:
842:
839:
836:
833:
809:
806:
803:
772:
769:
766:
763:
760:
740:
737:
734:
731:
728:
704:
701:
698:
678:
675:
672:
659:
658:
655:
652:
649:
643:
642:
639:
636:
633:
627:
626:
623:
620:
617:
611:
610:
607:
604:
601:
579:
576:
573:
570:
565:
561:
557:
552:
548:
515:
512:
509:
474:
459:
455:
451:
448:
445:
440:
436:
426:, hoping that
419:
416:
389:
388:
375:
370:
365:
361:
358:
355:
349:
344:
339:
334:
329:
325:
322:
319:
313:
308:
305:
278:
275:
272:
269:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
206:factorable as
200:
199:
188:
183:
179:
175:
170:
166:
162:
159:
132:, named after
125:
124:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3481:
3470:
3467:
3466:
3464:
3448:
3445:
3444:
3441:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3414:
3410:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3382:
3380:
3376:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3354:Pocklington's
3352:
3350:
3347:
3346:
3344:
3342:
3338:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3313:
3311:
3309:
3305:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3270:
3268:
3266:
3262:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3217:
3215:
3213:
3210:
3206:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3171:
3169:
3167:
3163:
3157:
3154:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3131:
3127:
3125:
3122:
3120:
3117:
3115:
3113:
3109:
3107:
3105:
3101:
3099:
3098:Pollard's rho
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3075:
3073:
3071:
3067:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3037:
3035:
3033:
3029:
3023:
3020:
3018:
3015:
3013:
3010:
3008:
3007:
3003:
3001:
3000:
2996:
2994:
2993:
2989:
2987:
2986:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2946:
2944:
2942:
2938:
2934:
2931:
2924:
2919:
2917:
2912:
2910:
2905:
2904:
2901:
2894:
2891:
2888:
2885:
2884:
2880:
2873:
2868:
2864:
2860:
2856:
2851:
2847:
2842:
2841:
2837:
2828:
2824:
2819:
2814:
2810:
2806:
2805:
2797:
2790:
2787:
2780:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2750:Factorization
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2712:
2708:
2706:
2704:
2700:
2696:
2680:
2677:
2672:
2668:
2659:
2655:
2651:
2643:
2641:
2622:
2618:
2614:
2610:
2603:
2595:
2579:
2575:
2571:
2562:
2560:
2556:
2552:
2548:
2532:
2529:
2523:
2520:
2517:
2514:
2488:
2485:
2479:
2476:
2473:
2470:
2444:
2441:
2421:
2418:
2410:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2350:
2346:
2342:
2335:
2319:
2315:
2311:
2302:
2300:
2292:
2290:
2288:
2284:
2280:
2274:
2270:
2266:
2262:
2258:
2253:
2251:
2246:
2238:
2231:
2229:
2228:
2224:
2207:
2203:
2194:
2192:
2191:
2187:
2184:
2182:
2181:
2177:
2174:
2171:
2170:
2166:
2159:
2157:
2156:
2152:
2135:
2131:
2122:
2120:
2119:
2115:
2112:
2110:
2109:
2105:
2102:
2099:
2098:
2095:
2081:
2078:
2075:
2066:
2046:
2042:
2015:
2011:
1990:
1987:
1982:
1978:
1965:
1945:
1942:
1937:
1933:
1918:
1915:
1912:
1909:
1907:
1904:
1903:
1899:
1896:
1893:
1890:
1888:
1885:
1884:
1880:
1877:
1874:
1871:
1869:
1866:
1865:
1862:
1861:are squares:
1846:
1842:
1821:
1818:
1815:
1803:
1796:
1793:
1791:
1787:
1784:
1783:
1779:
1776:
1774:
1771:
1770:
1766:
1764:1,254,441,084
1763:
1761:
1758:
1757:
1753:
1750:
1748:
1745:
1744:
1741:
1738:
1724:
1721:
1704:
1683:
1680:
1663:
1640:
1637:
1632:
1615:
1609:
1592:
1557:
1534:
1510:
1505:
1488:
1478:
1475:
1461:
1458:
1455:
1452:
1449:
1441:
1437:
1427:
1424:
1421:
1418:
1416:
1412:
1409:
1408:
1404:
1401:
1398:
1395:
1393:
1390:
1389:
1385:
1382:
1379:
1376:
1374:
1371:
1370:
1366:
1363:
1360:
1357:
1355:
1352:
1351:
1347:
1344:
1341:
1338:
1335:
1334:
1331:
1315:
1304:
1300:
1295:
1289:
1278:
1276:
1258:
1234:
1230:
1226:
1220:
1216:
1213:
1209:
1198:
1194:
1175:
1169:
1162:), one needs
1149:
1146:
1143:
1135:
1130:
1116:
1113:
1109:
1103:
1095:
1092:
1087:
1079:
1076:
1072:
1066:
1056:
1051:
1046:
1038:
1033:
1028:
1025:
1021:
1014:
1011:
1008:
985:
981:
974:
971:
968:
962:
959:
951:
935:
932:
929:
926:
917:
915:
899:
896:
893:
890:
887:
879:
860:
857:
854:
847:
843:
840:
837:
834:
831:
823:
807:
804:
801:
793:
789:
784:
770:
767:
764:
761:
758:
738:
735:
732:
729:
726:
702:
699:
696:
676:
673:
670:
656:
653:
650:
648:
645:
644:
640:
637:
634:
632:
629:
628:
624:
621:
618:
616:
613:
612:
608:
605:
602:
599:
598:
595:
593:
577:
574:
571:
568:
563:
559:
555:
550:
546:
529:
513:
510:
507:
497:
494:a - sqrt(b2)
493:
489:
486:
482:
478:
473:
457:
453:
449:
446:
443:
438:
434:
425:
417:
415:
412:
410:
406:
402:
398:
395:is odd, then
394:
373:
368:
363:
359:
356:
353:
347:
342:
337:
332:
327:
323:
320:
317:
311:
306:
303:
296:
295:
294:
292:
276:
273:
270:
267:
258:
256:
237:
234:
231:
222:
219:
216:
205:
204:algebraically
186:
181:
177:
173:
168:
164:
160:
157:
150:
149:
148:
146:
142:
139:
135:
131:
121:
118:
110:
107:February 2022
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: â
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
18:Fermat method
3446:
3140:
3128:
3111:
3103:
3022:MillerâRabin
3004:
2997:
2990:
2985:LucasâLehmer
2983:
2862:
2858:
2845:
2808:
2802:
2789:
2745:Prime factor
2702:
2698:
2694:
2647:
2593:
2563:
2558:
2554:
2550:
2546:
2408:
2303:
2298:
2296:
2286:
2282:
2276:
2272:
2268:
2264:
2260:
2256:
2249:
2247:
2244:
2067:
1924:
1905:
1886:
1867:
1807:
1789:
1785:
1772:
1759:
1746:
1739:
1479:
1476:
1439:
1435:
1433:
1414:
1410:
1391:
1372:
1353:
1302:
1298:
1293:
1287:
1284:
1274:
1196:
1192:
1133:
1131:
949:
918:
913:
877:
821:
791:
787:
785:
662:
646:
630:
614:
591:
527:
499:
495:
491:
488:
484:
481:repeat until
480:
476:
472:, a square.
423:
421:
418:Basic method
413:
408:
404:
400:
396:
392:
390:
290:
259:
254:
201:
129:
128:
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
3278:Pollard rho
3235:Goldschmidt
2969:Pocklington
2959:BaillieâPSW
2222:can only be
2175:Squares are
2150:can only be
2113:N mod 16 is
2103:Squares are
794:. That is,
3390:Cornacchia
3385:Chakravala
2933:algorithms
2838:References
2658:semiprimes
2545:. (Unless
2185:N mod 9 is
2100:modulo 16:
2082:2345678917
1822:2345678917
916:is prime.
411:be even.)
77:newspapers
3364:Berlekamp
3321:Euclidean
3209:Euclidean
3189:ToomâCook
3184:Karatsuba
2730:FOIL rule
2678:−
2389:⋅
2279:recursion
2172:modulo 9:
1988:−
1943:−
1797:24,582.2
1780:35,419.8
1638:−
1610:−
1453:−
1428:47,830.1
1093:−
1052:−
1029:−
897:⋅
835:−
824:, and so
730:−
569:−
444:−
357:−
343:−
235:−
174:−
3463:Category
3331:Lehmer's
3225:Chunking
3212:division
3141:Fermat's
2709:See also
2557:divides
2549:divides
2277:But the
2259:modulus
1900:367,179
1794:24,582.9
1777:35,418.1
1425:47,915.1
1422:48,017.5
1419:48,156.3
1386:367,179
3447:Italics
3369:Kunerth
3349:Cipolla
3230:Fourier
3199:FĂŒrer's
3093:Euler's
3083:Dixon's
3006:PĂ©pin's
2827:2005940
2695:product
2236:must be
2164:must be
1897:270,308
1894:173,439
1881:48,436
1754:60,002
1462:47830.1
1383:270,308
1380:173,439
1367:48,436
538:. Then
293:, then
143:as the
141:integer
91:scholar
3429:Schoof
3316:Binary
3220:Binary
3156:Shor's
2974:Fermat
2825:
2640:time.
1964:modulo
1919:605.9
1891:76,572
1878:48,435
1875:48,434
1872:48,433
1751:60,001
1405:605.9
1377:76,572
1364:48,435
1361:48,434
1358:48,433
948:, let
492:return
391:Since
93:
86:
79:
72:
64:
3250:Short
2979:Lucas
2823:JSTOR
2799:(PDF)
2781:Notes
2273:enddo
2269:endif
2261:times
1916:519.9
1913:416.5
1910:276.7
1725:55000
1684:48436
1402:519.9
1399:416.5
1396:276.7
1249:from
654:16.79
651:11.18
600:Try:
98:JSTOR
84:books
3245:Long
3179:Long
2652:and
2501:and
2434:and
2232:and
2160:and
1549:and
1506:>
1348:4th
1336:Try
1296:and
919:For
790:and
751:and
719:are
717:5959
641:441
572:5959
532:5959
514:5959
407:and
399:and
70:news
3409:LLL
3255:SRT
3114:+ 1
3106:â 1
2954:APR
2949:AKS
2867:doi
2813:doi
2594:Nuv
2561:.)
2553:or
2509:gcd
2465:gcd
2409:Nuv
2195:so
2123:so
1345:3rd
1342:2nd
1339:1st
1132:If
771:101
657:21
638:282
635:125
625:80
578:125
483:b2
138:odd
53:by
3465::
3413:KZ
3411:;
2863:68
2861:.
2857:.
2821:.
2809:28
2807:.
2801:.
2705:.
2301:.
2287:a'
2265:if
2257:do
2225:7
2188:7
2153:9
2116:5
2094:,
1788:â
1413:â
1301:â
1277:.
1129:.
783:.
739:59
703:21
689:,
677:80
622:79
619:78
609:3
560:78
536:78
485:is
257:.
147::
3415:)
3407:(
3112:p
3104:p
2922:e
2915:t
2908:v
2875:.
2869::
2829:.
2815::
2703:n
2699:n
2681:n
2673:2
2669:a
2628:)
2623:3
2619:/
2615:1
2611:N
2607:(
2604:O
2580:v
2576:/
2572:u
2559:v
2555:d
2551:u
2547:c
2533:d
2530:=
2527:)
2524:u
2521:d
2518:,
2515:N
2512:(
2489:c
2486:=
2483:)
2480:v
2477:c
2474:,
2471:N
2468:(
2445:u
2442:d
2422:v
2419:c
2395:u
2392:d
2386:v
2383:c
2380:=
2377:v
2374:u
2371:N
2351:u
2347:/
2343:v
2320:c
2316:/
2312:d
2299:N
2283:a
2250:a
2234:a
2208:2
2204:a
2162:a
2136:2
2132:a
2079:=
2076:N
2063:b
2047:2
2043:b
2032:a
2016:2
2012:a
1991:N
1983:2
1979:a
1968:a
1960:a
1946:N
1938:2
1934:a
1906:b
1887:b
1868:a
1847:2
1843:b
1819:=
1816:N
1790:b
1786:a
1773:b
1760:b
1747:a
1722:=
1716:x
1713:a
1710:m
1705:a
1681:=
1675:x
1672:a
1669:m
1664:a
1641:N
1633:2
1627:x
1624:a
1621:m
1616:a
1604:x
1601:a
1598:m
1593:a
1569:x
1566:a
1563:m
1558:a
1535:N
1511:N
1500:x
1497:a
1494:m
1489:a
1459:=
1456:b
1450:a
1440:N
1436:b
1415:b
1411:a
1392:b
1373:b
1354:a
1316:N
1303:b
1299:a
1294:b
1288:N
1275:N
1259:N
1235:4
1231:/
1227:1
1221:)
1217:N
1214:4
1210:(
1197:c
1193:N
1179:)
1176:N
1173:(
1170:O
1150:1
1147:=
1144:c
1134:N
1117:c
1114:2
1110:/
1104:2
1100:)
1096:c
1088:N
1083:(
1080:=
1077:2
1073:/
1067:2
1063:)
1057:c
1047:d
1042:(
1039:=
1034:N
1026:2
1022:/
1018:)
1015:d
1012:+
1009:c
1006:(
986:2
982:/
978:)
975:d
972:+
969:c
966:(
963:=
960:a
950:c
936:d
933:c
930:=
927:N
914:N
900:N
894:1
891:=
888:N
878:N
864:)
861:b
858:+
855:a
852:(
848:/
844:N
841:=
838:b
832:a
822:N
808:b
805:+
802:a
792:b
788:a
768:=
765:b
762:+
759:a
736:=
733:b
727:a
700:=
697:b
674:=
671:a
647:b
631:b
615:a
606:2
603:1
592:a
575:=
564:2
556:=
551:2
547:b
528:a
511:=
508:N
458:2
454:b
450:=
447:N
439:2
435:a
424:a
409:d
405:c
401:d
397:c
393:N
374:2
369:)
364:2
360:d
354:c
348:(
338:2
333:)
328:2
324:d
321:+
318:c
312:(
307:=
304:N
291:N
277:d
274:c
271:=
268:N
255:N
241:)
238:b
232:a
229:(
226:)
223:b
220:+
217:a
214:(
187:.
182:2
178:b
169:2
165:a
161:=
158:N
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
47:.
20:)
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