1490:
491:
892:
788:
397:
187:
334:
1494:
434:
1002:
703:
965:
78:
736:
110:
536:
1035:
218:
1359:
995:
917:
1227:
1550:
1169:
988:
1098:
1275:
1073:
1184:
1222:
1159:
1103:
1066:
1364:
1255:
1174:
1164:
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1440:
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439:
1407:
817:
1321:
741:
339:
17:
1486:
1476:
1435:
1211:
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1179:
1050:
220:
exist, there can be no primitive solution to the original equation. Without loss of generality, we can assume that
135:
283:
1412:
545:
until either a solution is found or all roots have been exhausted. In this case there is no primitive solution.
1374:
1247:
1093:
1045:
594:, then all solutions are primitive). Thus the above algorithm can be used to search for a primitive solution
1389:
1280:
1500:
1450:
1430:
402:
1151:
1126:
1055:
1510:
705:. A square root of −6 (mod 103) is 32, and 103 ⥠7 (mod 32); since
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1505:
1397:
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1354:
1316:
1060:
29:
921:
34:
1515:
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1402:
1306:
1265:
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708:
83:
1346:
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1131:
1030:
496:
190:
1087:
1080:
1466:
1422:
1136:
1113:
1311:
814:
Cornacchia, G. (1908). "Su di un metodo per la risoluzione in numeri interi dell' equazione
196:
1301:
1200:
1331:
1232:
1217:
1121:
1022:
1544:
1326:
1011:
1336:
980:
1014:
25:
121:
984:
124:. The algorithm was described in 1908 by Giuseppe Cornacchia.
576:, note that the existence of such a solution implies that
1531:
indicate that algorithm is for numbers of special forms
747:
451:
924:
820:
744:
711:
662:
499:
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405:
342:
286:
199:
138:
86:
37:
1459:
1421:
1388:
1345:
1289:
1246:
1150:
1112:
1021:
486:{\displaystyle s={\sqrt {\tfrac {m-r_{k}^{2}}{d}}}}
959:
887:{\displaystyle \sum _{h=0}^{n}C_{h}x^{n-h}y^{h}=P}
886:
782:
730:
697:
530:
485:
428:
391:
328:
212:
181:
104:
72:
783:{\displaystyle {\sqrt {\tfrac {103-7^{2}}{6}}}=3}
392:{\displaystyle r_{2}\equiv r_{0}{\pmod {r_{1}}}}
996:
648:will be a solution to the original equation.
182:{\displaystyle r_{0}^{2}\equiv -d{\pmod {m}}}
8:
329:{\displaystyle r_{1}\equiv m{\pmod {r_{0}}}}
1003:
989:
981:
945:
929:
923:
872:
856:
846:
836:
825:
819:
760:
745:
743:
716:
710:
683:
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661:
510:
498:
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303:
291:
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204:
198:
163:
148:
143:
137:
85:
58:
42:
36:
806:
189:(perhaps by using an algorithm listed
896:Giornale di Matematiche di Battaglini
636:. If such a solution is found, then
7:
493:is an integer, then the solution is
429:{\displaystyle r_{k}<{\sqrt {m}}}
374:
311:
171:
14:
1212:Special number field sieve (SNFS)
1206:General number field sieve (GNFS)
698:{\displaystyle x^{2}+6y^{2}=103}
548:To find non-primitive solutions
538:; otherwise try another root of
269:, which will still be a root of
367:
304:
164:
960:{\displaystyle x^{2}+dy^{2}=m}
385:
368:
322:
305:
175:
165:
73:{\displaystyle x^{2}+dy^{2}=m}
1:
1170:Lenstra elliptic curve (ECM)
731:{\displaystyle 7^{2}<103}
132:First, find any solution to
105:{\displaystyle 1\leq d<m}
1551:Number theoretic algorithms
586:(and equivalently, that if
531:{\displaystyle x=r_{k},y=s}
18:computational number theory
1567:
1477:Exponentiation by squaring
1160:Continued fraction (CFRAC)
1524:
1390:Greatest common divisor
916:Basilla, J. M. (2004).
1501:Modular exponentiation
961:
888:
841:
790:, there is a solution
784:
732:
699:
532:
487:
430:
393:
330:
247:(if not, then replace
214:
183:
106:
74:
22:Cornacchia's algorithm
1228:Shanks's square forms
1152:Integer factorization
1127:Sieve of Eratosthenes
962:
918:"On the solutions of
889:
821:
794: = 7,
785:
733:
700:
533:
488:
431:
399:and so on; stop when
394:
331:
215:
213:{\displaystyle r_{0}}
184:
107:
75:
1506:Montgomery reduction
1380:Function field sieve
1355:Baby-step giant-step
1201:Quadratic sieve (QS)
922:
818:
742:
709:
660:
497:
440:
403:
340:
284:
197:
136:
84:
35:
30:Diophantine equation
1516:Trachtenberg system
1482:Integer square root
1423:Modular square root
1142:Wheel factorization
1094:Quadratic Frobenius
1074:LucasâLehmerâRiesel
656:Solve the equation
474:
278:Euclidean algorithm
153:
1408:Extended Euclidean
1347:Discrete logarithm
1276:SchönhageâStrassen
1132:Sieve of Pritchard
957:
884:
780:
771:
728:
695:
528:
483:
480:
460:
426:
389:
326:
210:
179:
139:
102:
70:
1538:
1537:
1137:Sieve of Sundaram
772:
770:
481:
479:
424:
276:). Then use the
1558:
1487:Integer relation
1460:Other algorithms
1365:Pollard kangaroo
1256:Ancient Egyptian
1114:Prime-generating
1099:SolovayâStrassen
1012:Number-theoretic
1005:
998:
991:
982:
977:
974:Proc. Japan Acad
971:
966:
964:
963:
958:
950:
949:
934:
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904:
903:
893:
891:
890:
885:
877:
876:
867:
866:
851:
850:
840:
835:
811:
798: = 3.
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773:
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28:for solving the
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1520:
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1108:
1081:Proth's theorem
1023:Primality tests
1017:
1009:
976:. 80(A): 40â41.
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1448:
1443:
1441:TonelliâShanks
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1418:
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1415:
1410:
1405:
1400:
1394:
1392:
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1383:
1382:
1377:
1375:Index calculus
1372:
1370:PohligâHellman
1367:
1362:
1357:
1351:
1349:
1343:
1342:
1340:
1339:
1334:
1329:
1324:
1322:Newton-Raphson
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1252:
1250:
1248:Multiplication
1244:
1243:
1241:
1240:
1235:
1233:Trial division
1230:
1225:
1220:
1218:Rational sieve
1215:
1208:
1203:
1198:
1190:
1182:
1177:
1172:
1167:
1162:
1156:
1154:
1148:
1147:
1145:
1144:
1139:
1134:
1129:
1124:
1122:Sieve of Atkin
1118:
1116:
1110:
1109:
1107:
1106:
1101:
1096:
1091:
1084:
1077:
1070:
1063:
1058:
1053:
1048:
1046:Elliptic curve
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1038:
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910:External links
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193:); if no such
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1437:
1436:Pocklington's
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1207:
1204:
1202:
1199:
1197:
1195:
1191:
1189:
1187:
1183:
1181:
1180:Pollard's rho
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1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
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1155:
1153:
1149:
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1020:
1016:
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994:
992:
987:
986:
983:
975:
968:
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951:
946:
942:
938:
935:
930:
926:
914:
913:
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901:
897:
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878:
873:
869:
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860:
857:
853:
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843:
837:
832:
829:
826:
822:
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128:The algorithm
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125:
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119:
115:
99:
96:
93:
90:
87:
67:
64:
59:
55:
51:
48:
43:
39:
31:
27:
23:
19:
1528:
1471:
1210:
1193:
1185:
1104:MillerâRabin
1086:
1079:
1072:
1067:LucasâLehmer
1065:
973:
899:
895:
809:
795:
791:
655:
643:
639:
628:
620:
612:
608:
601:
597:
578:
571:
567:
563:
555:
551:
547:
541:
272:
262:
258:
249:
233:
222:
131:
117:
113:
21:
15:
1360:Pollard rho
1317:Goldschmidt
1051:Pocklington
1041:BaillieâPSW
592:square-free
1472:Cornacchia
1467:Chakravala
1015:algorithms
802:References
1446:Berlekamp
1403:Euclidean
1291:Euclidean
1271:ToomâCook
1266:Karatsuba
861:−
823:∑
754:−
458:−
354:≡
298:≡
158:−
155:≡
91:≤
26:algorithm
1545:Category
1413:Lehmer's
1307:Chunking
1294:division
1223:Fermat's
902:: 33â90.
582:divides
280:to find
80:, where
1529:Italics
1451:Kunerth
1431:Cipolla
1312:Fourier
1281:FĂŒrer's
1175:Euler's
1165:Dixon's
1088:PĂ©pin's
652:Example
633:
617:
244:
230:
122:coprime
1511:Schoof
1398:Binary
1302:Binary
1238:Shor's
1056:Fermat
560:where
24:is an
1332:Short
1061:Lucas
970:(PDF)
436:. If
256:with
1327:Long
1261:Long
738:and
723:<
570:) =
562:gcd(
417:<
191:here
120:are
116:and
112:and
97:<
1491:LLL
1337:SRT
1196:+ 1
1188:â 1
1036:APR
1031:AKS
894:".
751:103
726:103
693:103
606:to
590:is
574:â 1
372:mod
309:mod
169:mod
16:In
1547::
1495:KZ
1493:;
972:.
900:46
898:.
644:gv
642:,
640:gu
615:=
613:dv
611:+
600:,
566:,
554:,
336:,
261:-
228:â€
20:,
1497:)
1489:(
1194:p
1186:p
1004:e
997:t
990:v
967:"
955:m
952:=
947:2
943:y
939:d
936:+
931:2
927:x
882:P
879:=
874:h
870:y
864:h
858:n
854:x
848:h
844:C
838:n
833:0
830:=
827:h
796:y
792:x
778:3
775:=
768:6
762:2
758:7
718:2
714:7
690:=
685:2
681:y
677:6
674:+
669:2
665:x
646:)
638:(
629:g
625:/
621:m
609:u
604:)
602:v
598:u
596:(
588:m
584:m
579:g
572:g
568:y
564:x
558:)
556:y
552:x
550:(
542:d
540:-
526:s
523:=
520:y
517:,
512:k
508:r
504:=
501:x
477:d
471:2
466:k
462:r
455:m
447:=
444:s
422:m
412:k
408:r
386:)
381:1
377:r
369:(
362:0
358:r
349:2
345:r
323:)
318:0
314:r
306:(
301:m
293:1
289:r
273:d
271:-
266:0
263:r
259:m
253:0
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241:2
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234:m
226:0
223:r
206:0
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173:m
166:(
161:d
150:2
145:0
141:r
118:m
114:d
100:m
94:d
88:1
68:m
65:=
60:2
56:y
52:d
49:+
44:2
40:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
