293:
1006:" or "F4". This value was used in RSA mainly for historical reasons; early raw RSA implementations (without proper padding) were vulnerable to very small exponents, while use of high exponents was computationally expensive with no advantage to security (assuming proper padding).
838:
924:
193:
753:
687:
1017:, which ensures that any seed value will be coprime to it (vital to ensure the maximum period) while also allowing efficient reduction by the modulus using a bit shift and subtract.
618:
549:
480:
361:
967:
387:
843:
766:
858:
1092:
1067:
163:
1134:
1051:
693:
1010:
17:
624:
555:
486:
417:
1120:
929:
404:
gave the first explicit construction of this polygon. In number theory, primes of this form are known as
326:
939:
226:
1138:
1043:
397:
301:
401:
848:
8:
1096:
1071:
366:
292:
1174:
1047:
1036:
409:
112:
982:
390:
122:
71:
58:
970:
758:
312:
200:
156:
37:
1168:
1142:
405:
213:
134:
1108:-F4|-3 the public exponent to use, either 65537 or 3. The default is 65537.
146:
139:
1014:
278:
265:
1156:
The ZX Spectrum uses p=65537 and a=75, and stores some bi-1 in memory.
393:
297:
833:{\displaystyle 2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417}
104:
63:
919:{\displaystyle 2^{2^{6}}+1=2^{64}+1=274177\times 67280421310721}
239:
101:
98:
95:
92:
89:
86:
83:
80:
77:
252:
1147:(2nd ed.). Sinclair Research Ltd. pp. 73â75
942:
861:
769:
696:
627:
558:
489:
420:
369:
329:
188:{\displaystyle {\stackrel {\digamma }{\mathrm {M} }}}
166:
74:
323:
65537 is the largest known prime number of the form
981:65537 is commonly used as a public exponent in the
1035:
961:
918:
832:
747:
681:
612:
543:
474:
381:
355:
187:
129:(sixty-five thousand five hundred thirty-seventh)
1166:
761:found that the next Fermat number is composite:
985:cryptosystem. Because it is the Fermat number
117:sixty-five thousand five hundred thirty-seven
748:{\displaystyle 2^{2^{4}}+1=2^{16}+1=65537.}
1009:65537 is also used as the modulus in some
932:, and currently the largest known integer
412:. The only known prime Fermat numbers are
1033:
682:{\displaystyle 2^{2^{3}}+1=2^{8}+1=257,}
400:with compass and unmarked straightedge.
291:
1133:
613:{\displaystyle 2^{2^{2}}+1=2^{4}+1=17,}
14:
1167:
1144:Sinclair ZX Spectrum Basic Programming
544:{\displaystyle 2^{2^{1}}+1=2^{2}+1=5,}
475:{\displaystyle 2^{2^{0}}+1=2^{1}+1=3,}
1042:. New York: Springer-Verlag. p.
1068:"Sequences by difficulty of search"
24:
1034:Conway, J. H.; Guy, R. K. (1996).
172:
25:
1186:
1095:. OpenSSL Project. Archived from
318:
408:, named after the mathematician
1011:Lehmer random number generators
976:
1127:
1113:
1085:
1060:
1027:
13:
1:
1020:
1139:"Chapter 11. Random numbers"
1002:, the common shorthand is "F
7:
1121:"RSA with small exponents?"
356:{\displaystyle 2^{2^{n}}+1}
10:
1191:
1013:, such as the one used by
962:{\displaystyle 10^{n}+27}
277:
264:
251:
238:
225:
212:
199:
155:
145:
133:
121:
111:
53:
32:
394:polygon with 65537 sides
930:JacobsthalâLucas number
928:65537 is also the 17th
963:
920:
834:
749:
683:
614:
545:
476:
383:
357:
305:
189:
964:
936:for which the number
921:
835:
750:
684:
615:
546:
477:
384:
358:
311:is the integer after
302:constructible polygon
295:
190:
940:
859:
767:
694:
625:
556:
487:
418:
402:Johann Gustav Hermes
367:
327:
164:
1038:The Book of Numbers
382:{\displaystyle n=4}
959:
916:
830:
745:
679:
610:
541:
472:
379:
353:
315:and before 65538.
306:
296:Construction of a
185:
298:regular 65537-gon
290:
289:
218:10000000000000001
182:
49:
48:
16:(Redirected from
1182:
1159:
1158:
1153:
1152:
1131:
1125:
1124:
1117:
1111:
1110:
1105:
1104:
1089:
1083:
1082:
1080:
1079:
1070:. Archived from
1064:
1058:
1057:
1041:
1031:
1001:
994:
968:
966:
965:
960:
952:
951:
925:
923:
922:
917:
897:
896:
878:
877:
876:
875:
852:
839:
837:
836:
831:
805:
804:
786:
785:
784:
783:
754:
752:
751:
746:
732:
731:
713:
712:
711:
710:
688:
686:
685:
680:
663:
662:
644:
643:
642:
641:
619:
617:
616:
611:
594:
593:
575:
574:
573:
572:
550:
548:
547:
542:
525:
524:
506:
505:
504:
503:
481:
479:
478:
473:
456:
455:
437:
436:
435:
434:
410:Pierre de Fermat
389:). Therefore, a
388:
386:
385:
380:
362:
360:
359:
354:
346:
345:
344:
343:
207:
194:
192:
191:
186:
184:
183:
181:
176:
175:
169:
34:
33:
30:
29:
21:
1190:
1189:
1185:
1184:
1183:
1181:
1180:
1179:
1165:
1164:
1163:
1162:
1150:
1148:
1132:
1128:
1119:
1118:
1114:
1102:
1100:
1091:
1090:
1086:
1077:
1075:
1066:
1065:
1061:
1054:
1032:
1028:
1023:
1005:
996:
992:
986:
979:
947:
943:
941:
938:
937:
892:
888:
871:
867:
866:
862:
860:
857:
856:
846:
800:
796:
779:
775:
774:
770:
768:
765:
764:
727:
723:
706:
702:
701:
697:
695:
692:
691:
658:
654:
637:
633:
632:
628:
626:
623:
622:
589:
585:
568:
564:
563:
559:
557:
554:
553:
520:
516:
499:
495:
494:
490:
488:
485:
484:
451:
447:
430:
426:
425:
421:
419:
416:
415:
368:
365:
364:
339:
335:
334:
330:
328:
325:
324:
321:
286:
273:
260:
247:
234:
221:
205:
177:
171:
170:
168:
167:
165:
162:
161:
128:
107:
69:
68:
59:List of numbers
28:
23:
22:
15:
12:
11:
5:
1188:
1178:
1177:
1161:
1160:
1135:Vickers, Steve
1126:
1112:
1084:
1059:
1052:
1025:
1024:
1022:
1019:
1003:
988:
978:
975:
971:probable prime
958:
955:
950:
946:
915:
914:67280421310721
912:
909:
906:
903:
900:
895:
891:
887:
884:
881:
874:
870:
865:
844:Fortuné Landry
829:
826:
823:
820:
817:
814:
811:
808:
803:
799:
795:
792:
789:
782:
778:
773:
759:Leonhard Euler
744:
741:
738:
735:
730:
726:
722:
719:
716:
709:
705:
700:
678:
675:
672:
669:
666:
661:
657:
653:
650:
647:
640:
636:
631:
609:
606:
603:
600:
597:
592:
588:
584:
581:
578:
571:
567:
562:
540:
537:
534:
531:
528:
523:
519:
515:
512:
509:
502:
498:
493:
471:
468:
465:
462:
459:
454:
450:
446:
443:
440:
433:
429:
424:
378:
375:
372:
352:
349:
342:
338:
333:
320:
319:In mathematics
317:
288:
287:
284:
281:
275:
274:
271:
268:
262:
261:
258:
255:
249:
248:
245:
242:
236:
235:
232:
229:
223:
222:
219:
216:
210:
209:
203:
197:
196:
180:
174:
159:
153:
152:
149:
143:
142:
137:
131:
130:
125:
119:
118:
115:
109:
108:
70:
67:
66:
61:
55:
54:
51:
50:
47:
46:
43:
40:
38:← 65536
27:Natural number
26:
18:65537 (number)
9:
6:
4:
3:
2:
1187:
1176:
1173:
1172:
1170:
1157:
1146:
1145:
1140:
1136:
1130:
1122:
1116:
1109:
1099:on 2017-03-13
1098:
1094:
1088:
1074:on 2014-07-14
1073:
1069:
1063:
1055:
1053:0-387-97993-X
1049:
1045:
1040:
1039:
1030:
1026:
1018:
1016:
1012:
1007:
999:
991:
984:
974:
972:
956:
953:
948:
944:
935:
931:
926:
913:
910:
907:
904:
901:
898:
893:
889:
885:
882:
879:
872:
868:
863:
854:
850:
845:
840:
827:
824:
821:
818:
815:
812:
809:
806:
801:
797:
793:
790:
787:
780:
776:
771:
762:
760:
755:
742:
739:
736:
733:
728:
724:
720:
717:
714:
707:
703:
698:
689:
676:
673:
670:
667:
664:
659:
655:
651:
648:
645:
638:
634:
629:
620:
607:
604:
601:
598:
595:
590:
586:
582:
579:
576:
569:
565:
560:
551:
538:
535:
532:
529:
526:
521:
517:
513:
510:
507:
500:
496:
491:
482:
469:
466:
463:
460:
457:
452:
448:
444:
441:
438:
431:
427:
422:
413:
411:
407:
406:Fermat primes
403:
399:
398:constructible
395:
392:
376:
373:
370:
350:
347:
340:
336:
331:
316:
314:
310:
303:
299:
294:
282:
280:
276:
269:
267:
263:
256:
254:
250:
243:
241:
237:
230:
228:
224:
217:
215:
211:
204:
202:
201:Roman numeral
198:
178:
160:
158:
157:Greek numeral
154:
150:
148:
144:
141:
138:
136:
135:Factorization
132:
126:
124:
120:
116:
114:
110:
106:
103:
100:
97:
94:
91:
88:
85:
82:
79:
76:
73:
65:
62:
60:
57:
56:
52:
45:65538 →
44:
41:
39:
36:
35:
31:
19:
1155:
1149:. Retrieved
1143:
1129:
1115:
1107:
1101:. Retrieved
1097:the original
1087:
1076:. Retrieved
1072:the original
1062:
1037:
1029:
1008:
997:
989:
980:
977:Applications
933:
927:
855:
853:showed that
841:
763:
756:
690:
621:
552:
483:
414:
322:
308:
307:
1093:"genrsa(1)"
1015:ZX Spectrum
847: [
279:Hexadecimal
231:10022220022
1151:2022-05-26
1103:2017-05-24
1078:2014-06-14
1021:References
816:4294967297
266:Duodecimal
911:×
842:In 1880,
825:×
757:In 1732,
179:ϝ
1175:Integers
1169:Category
1137:(1983).
113:Cardinal
64:Integers
993:= 2 + 1
828:6700417
391:regular
244:1223225
227:Ternary
208:DXXXVII
151:6,543rd
127:65537th
123:Ordinal
1050:
908:274177
743:65537.
300:. See
257:200001
240:Senary
214:Binary
195:͔ΔÏλζŽ
995:with
969:is a
851:]
313:65536
309:65537
283:10001
270:31B15
253:Octal
147:Prime
140:prime
42:65537
1048:ISBN
1044:139
1000:= 4
983:RSA
822:641
674:257
396:is
206:LXV
102:90k
99:80k
96:70k
93:60k
90:50k
87:40k
84:30k
81:20k
78:10k
1171::
1154:.
1141:.
1106:.
1046:.
973:.
957:27
945:10
894:64
849:fr
802:32
729:16
605:17
285:16
272:12
1123:.
1081:.
1056:.
1004:4
998:n
990:n
987:F
954:+
949:n
934:n
905:=
902:1
899:+
890:2
886:=
883:1
880:+
873:6
869:2
864:2
819:=
813:=
810:1
807:+
798:2
794:=
791:1
788:+
781:5
777:2
772:2
740:=
737:1
734:+
725:2
721:=
718:1
715:+
708:4
704:2
699:2
677:,
671:=
668:1
665:+
660:8
656:2
652:=
649:1
646:+
639:3
635:2
630:2
608:,
602:=
599:1
596:+
591:4
587:2
583:=
580:1
577:+
570:2
566:2
561:2
539:,
536:5
533:=
530:1
527:+
522:2
518:2
514:=
511:1
508:+
501:1
497:2
492:2
470:,
467:3
464:=
461:1
458:+
453:1
449:2
445:=
442:1
439:+
432:0
428:2
423:2
377:4
374:=
371:n
363:(
351:1
348:+
341:n
337:2
332:2
304:.
259:8
246:6
233:3
220:2
173:M
105:â
75:0
72:â
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
