32:
198:
is more likely to be divisible by two than by three, and so on. With this ordering, there is no point in testing for divisibility by four if the number has already been determined not divisible by two, and so on for three and any multiple of three, etc. Therefore, the effort can be reduced by
743:
to obtain the prime numbers as candidate factors. A useful table need not be large: P(3512) = 32749, the last prime that fits into a sixteen-bit signed integer and P(6542) = 65521 for unsigned sixteen-bit integers. That would suffice to test primality for numbers up to 65537 = 4,295,098,369.
697:
891:
are chosen to have large prime factors of similar size so that they cannot be factored by any publicly known method in a useful time period on any available computer system or computer cluster such as
748:) would only be worthwhile if many numbers were to be tested. If instead a variant is used without primality testing, but simply dividing by every odd number less than the square root the base-2
1570:
871:
However, many-digit numbers that do not have factors in the small primes can require days or months to factor with the trial division. In such cases other methods are used such as the
227:
812:, and so on. It can be shown that 88% of all positive integers have a factor under 100 and that 92% have a factor under 1000. Thus, when confronted by an arbitrary large
792:
729:
840:
866:
1574:
1082:
604:
1115:
61:
1052:
20:
1630:
1439:
1075:
800:
Even so, this is a quite satisfactory method, considering that even the best-known algorithms have exponential time growth. For
1307:
979:
1249:
1068:
574:, not just primes. A more complicated implementation only testing primes would be significantly faster in the worst case.
1178:
1355:
967:
1017:
83:
903:, a 250-digit number, using the GNFS and resources of several supercomputers. The running time was 2700 core years.
54:
19:
This article is about the mathematical algorithm. For the judicial chamber of the
International Criminal Court, see
1302:
1264:
1239:
1153:
1635:
1444:
1335:
1254:
1244:
1183:
1146:
1272:
1120:
1041:
206:
1525:
359:
1520:
1487:
1449:
1350:
920:
Mollin, Richard A. (2002). "A brief history of factoring and primality testing B. C. (before computers)".
358:
An example of the trial division algorithm, using successive integers as trial factors, is as follows (in
1401:
804:
chosen uniformly at random from integers of a given length, there is a 50% chance that 2 is a factor of
1566:
1556:
1515:
1291:
1285:
1259:
1130:
876:
190:
is divisible by any smaller number. Clearly, it is only worthwhile to test candidate factors less than
186:
refers to "the integer to be factored"), the trial division consists of systematically testing whether
1551:
1492:
44:
1454:
1327:
1173:
1125:
48:
40:
1469:
884:
732:
1360:
1580:
1530:
1510:
65:
1590:
1231:
1206:
1135:
762:
745:
705:
100:
819:
397:"""Return a list of the prime factors for a natural number."""
1585:
1477:
1459:
1434:
1396:
1140:
949:
1027:
989:
845:
107:, the integer to be factored, can be divided by each number in turn that is less than the
8:
1595:
1561:
1482:
1386:
1345:
1340:
1317:
1221:
879:(GNFS). Because these methods also have superpolynomial time growth a practical limit of
1426:
1373:
1370:
1211:
1160:
1110:
1046:
937:
1167:
1546:
1502:
1216:
1193:
1013:
975:
740:
692:{\displaystyle \pi (2^{n/2})\approx {2^{n/2} \over \left({\frac {n}{2}}\right)\ln 2}}
1042:
Wikiversity offers a lesson on prime factorization using trial division with Python.
797:
In both cases, the required time grows exponentially with the digits of the number.
1391:
1023:
997:
985:
929:
1381:
1280:
1009:
971:
945:
872:
103:
algorithms. The essential idea behind trial division tests to see if an integer
1411:
1297:
1201:
1102:
1001:
896:
1060:
1624:
1406:
1091:
892:
352:
203:
as candidate factors. Furthermore, the trial factors need go no further than
816:, it is worthwhile to check for divisibility by the small primes, since for
1416:
296:= 5, etc. Then the last prime number worth testing as a possible factor of
200:
166:
108:
941:
583:
339:= 48 not just 25 because the square of the next prime is 49, and below
1094:
161:
933:
899:. The largest cryptography-grade number that has been factored is
900:
594:, if it starts from two and works up only to the square root of
1047:
Fast JavaScript Prime Factor
Calculator using trial division
343:= 25 just 2 and 3 are sufficient. Should the square root of
335:
is a factor. Thus, testing with 2, 3, and 5 suffices up to
264:
A definite bound on the prime factors is possible. Suppose
570:
This version tests every integer up to the square root of
586:, trial division is a laborious algorithm. For a base-2
466:# The remainder of n divided by f might be zero.
1611:
indicate that algorithm is for numbers of special forms
253:
would have been detected earlier as being divisible by
99:
is the most laborious but easiest to understand of the
210:
848:
822:
765:
708:
607:
565:# Prime factors may be repeated: 12 factors to 2,2,3.
209:
194:, and in order from two upwards because an arbitrary
883:
digits is reached very quickly. For this reason, in
442:# While f is no larger than the square root of n ...
860:
834:
786:
739:. This does not take into account the overhead of
723:
691:
221:
996:
1622:
53:but its sources remain unclear because it lacks
1090:
1076:
1083:
1069:
1006:Prime numbers. A computational perspective
118:For example, to find the prime factors of
21:Judges of the International Criminal Court
964:A concrete introduction to higher algebra
756:, prime or not, it can take up to about:
487:# If so, it divides n. Add f to the list.
84:Learn how and when to remove this message
1053:Trial Division in Java, C and JavaScript
744:Preparing such a table (usually via the
222:{\displaystyle \scriptstyle {\sqrt {n}}}
808:and a 33% chance that 3 is a factor of
556:# Add the remaining prime n to the list
347:be an integer, then it is a factor and
1623:
961:
919:
160:Trial division was first described by
1064:
25:
13:
1049:. Can handle numbers up to about 2
968:Undergraduate Texts in Mathematics
14:
1647:
1292:Special number field sieve (SNFS)
1286:General number field sieve (GNFS)
1035:
735:, the number of primes less than
1631:Integer factorization algorithms
508:# But if f is not a factor of n,
418:# The first possible factor.
323:; equality here would mean that
30:
16:Integer factorization algorithm
1008:(2nd ed.). New York, NY:
970:(3rd ed.). New York, NY:
913:
718:
712:
632:
611:
499:# Divide that factor out of n.
1:
906:
520:# Add one to f and try again.
129:by successive primes: first,
1250:Lenstra elliptic curve (ECM)
233:is divisible by some number
7:
962:Childs, Lindsay N. (2009).
10:
1652:
1557:Exponentiation by squaring
1240:Continued fraction (CFRAC)
877:general number field sieve
18:
1604:
1539:
1501:
1468:
1425:
1369:
1326:
1230:
1192:
1101:
598:, the algorithm requires
173:
577:
406:# Prepare an empty list.
364:
257:or by a prime factor of
125:, one can try to divide
39:This article includes a
1470:Greatest common divisor
885:public key cryptography
787:{\displaystyle 2^{n/2}}
733:prime-counting function
724:{\displaystyle \pi (x)}
702:trial divisions, where
68:more precise citations.
1636:Division (mathematics)
1581:Modular exponentiation
862:
836:
835:{\displaystyle a=1000}
788:
725:
693:
223:
1308:Shanks's square forms
1232:Integer factorization
1207:Sieve of Eratosthenes
863:
837:
789:
746:Sieve of Eratosthenes
726:
694:
224:
101:integer factorization
1586:Montgomery reduction
1460:Function field sieve
1435:Baby-step giant-step
1281:Quadratic sieve (QS)
922:Mathematics Magazine
861:{\displaystyle n=10}
846:
820:
763:
706:
605:
207:
153:is itself prime. So
1596:Trachtenberg system
1562:Integer square root
1503:Modular square root
1222:Wheel factorization
1174:Quadratic Frobenius
1154:LucasâLehmerâRiesel
275:'th prime, so that
1488:Extended Euclidean
1427:Discrete logarithm
1356:SchönhageâStrassen
1212:Sieve of Pritchard
858:
832:
784:
721:
689:
245:were smaller than
219:
218:
41:list of references
1618:
1617:
1217:Sieve of Sundaram
998:Crandall, Richard
981:978-0-387-74527-5
741:primality testing
687:
671:
216:
178:Given an integer
94:
93:
86:
1643:
1567:Integer relation
1540:Other algorithms
1445:Pollard kangaroo
1336:Ancient Egyptian
1194:Prime-generating
1179:SolovayâStrassen
1092:Number-theoretic
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133:; next, neither
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89:
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78:
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69:
64:this article by
55:inline citations
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1161:Proth's theorem
1103:Primality tests
1097:
1089:
1056:(in Portuguese)
1055:
1038:
1020:
1010:Springer-Verlag
1002:Pomerance, Carl
982:
972:Springer-Verlag
958:
957:
934:10.2307/3219180
918:
914:
909:
873:quadratic sieve
847:
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199:selecting only
176:
154:
150:
146:
142:
141:evenly divides
138:
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126:
119:
112:
104:
90:
79:
73:
70:
59:
45:related reading
35:
31:
24:
17:
12:
11:
5:
1649:
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1528:
1523:
1521:TonelliâShanks
1518:
1513:
1507:
1505:
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1490:
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1472:
1466:
1465:
1463:
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1455:Index calculus
1452:
1450:PohligâHellman
1447:
1442:
1437:
1431:
1429:
1423:
1422:
1420:
1419:
1414:
1409:
1404:
1402:Newton-Raphson
1399:
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1358:
1353:
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1332:
1330:
1328:Multiplication
1324:
1323:
1321:
1320:
1315:
1313:Trial division
1310:
1305:
1300:
1298:Rational sieve
1295:
1288:
1283:
1278:
1270:
1262:
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1252:
1247:
1242:
1236:
1234:
1228:
1227:
1225:
1224:
1219:
1214:
1209:
1204:
1202:Sieve of Atkin
1198:
1196:
1190:
1189:
1187:
1186:
1181:
1176:
1171:
1164:
1157:
1150:
1143:
1138:
1133:
1128:
1126:Elliptic curve
1123:
1118:
1113:
1107:
1105:
1099:
1098:
1088:
1087:
1080:
1073:
1065:
1059:
1058:
1050:
1044:
1037:
1036:External links
1034:
1033:
1032:
1018:
994:
980:
956:
955:
911:
910:
908:
905:
897:computer grids
893:supercomputers
857:
854:
851:
831:
828:
825:
795:
794:
781:
777:
773:
769:
720:
717:
714:
711:
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675:
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655:
651:
647:
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637:
634:
629:
625:
621:
617:
613:
610:
579:
576:
370:trial_division
365:
353:perfect square
332: + 1
328:
316: + 1
312:
303:
293:
286:
279:
267:
215:
175:
172:
155:70 = 2 Ă 5 Ă 7
97:Trial division
92:
91:
49:external links
38:
36:
29:
15:
9:
6:
4:
3:
2:
1648:
1637:
1634:
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1610:
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1545:
1544:
1542:
1538:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1516:Pocklington's
1514:
1512:
1509:
1508:
1506:
1504:
1500:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
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1471:
1467:
1461:
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1453:
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1448:
1446:
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1287:
1284:
1282:
1279:
1277:
1275:
1271:
1269:
1267:
1263:
1261:
1260:Pollard's rho
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1237:
1235:
1233:
1229:
1223:
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1029:
1025:
1021:
1019:0-387-25282-7
1015:
1011:
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939:
935:
931:
927:
923:
916:
912:
904:
902:
898:
894:
890:
887:, values for
886:
882:
878:
874:
869:
855:
852:
849:
829:
826:
823:
815:
811:
807:
803:
798:
779:
775:
771:
767:
759:
758:
757:
755:
752:digit number
751:
747:
742:
738:
734:
715:
709:
683:
680:
677:
673:
668:
665:
660:
653:
649:
645:
641:
635:
627:
623:
619:
615:
608:
601:
600:
599:
597:
593:
590:digit number
589:
585:
575:
573:
363:
361:
356:
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346:
342:
338:
331:
327:
321:
315:
311:
299:
292:
285:
278:
262:
260:
256:
252:
248:
244:
240:
236:
232:
213:
202:
201:prime numbers
197:
193:
189:
185:
181:
171:
169:
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163:
158:
122:
116:
110:
102:
98:
88:
85:
77:
67:
63:
57:
56:
50:
46:
42:
37:
28:
27:
22:
1608:
1312:
1290:
1273:
1265:
1184:MillerâRabin
1166:
1159:
1152:
1147:LucasâLehmer
1145:
1005:
963:
928:(1): 18â29.
925:
921:
915:
888:
880:
870:
842:, in base-2
813:
809:
805:
801:
799:
796:
753:
749:
736:
731:denotes the
701:
595:
591:
587:
581:
571:
569:
357:
348:
344:
340:
336:
329:
325:
319:
313:
309:
297:
290:
283:
276:
263:
258:
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250:
246:
242:
238:
234:
230:
229:because, if
195:
191:
187:
183:
179:
177:
165:
164:in his book
159:
120:
117:
96:
95:
80:
71:
60:Please help
52:
1440:Pollard rho
1397:Goldschmidt
1131:Pocklington
1121:BaillieâPSW
167:Liber Abaci
145:; finally,
131:70 / 2 = 35
109:square root
66:introducing
1625:Categories
1552:Cornacchia
1547:Chakravala
1095:algorithms
1028:1088.11001
990:1165.00002
907:References
584:worst case
147:35 / 5 = 7
74:March 2014
1526:Berlekamp
1483:Euclidean
1371:Euclidean
1351:ToomâCook
1346:Karatsuba
710:π
681:
636:≈
609:π
239:n = p Ă q
162:Fibonacci
1493:Lehmer's
1387:Chunking
1374:division
1303:Fermat's
1004:(2005).
875:and the
170:(1202).
1609:Italics
1531:Kunerth
1511:Cipolla
1392:Fourier
1361:FĂŒrer's
1255:Euler's
1245:Dixon's
1168:PĂ©pin's
950:2107288
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