282:
995:" 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).
827:
913:
182:
742:
676:
1006:, 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.
607:
538:
469:
350:
956:
376:
832:
755:
847:
1081:
1056:
152:
1123:
1040:
682:
999:
613:
544:
475:
406:
1109:
918:
393:
gave the first explicit construction of this polygon. In number theory, primes of this form are known as
315:
928:
215:
1127:
1032:
386:
290:
390:
837:
8:
1085:
1060:
355:
281:
1163:
1036:
1025:
398:
101:
971:
379:
111:
60:
47:
959:
747:
301:
189:
145:
26:
1157:
1131:
394:
202:
123:
1097:-F4|-3 the public exponent to use, either 65537 or 3. The default is 65537.
135:
128:
1003:
267:
254:
1145:
The ZX Spectrum uses p=65537 and a=75, and stores some bi-1 in memory.
382:
286:
822:{\displaystyle 2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417}
93:
52:
908:{\displaystyle 2^{2^{6}}+1=2^{64}+1=274177\times 67280421310721}
228:
90:
87:
84:
81:
78:
75:
72:
69:
66:
241:
1136:(2nd ed.). Sinclair Research Ltd. pp. 73â75
931:
850:
758:
685:
616:
547:
478:
409:
358:
318:
177:{\displaystyle {\stackrel {\digamma }{\mathrm {M} }}}
155:
63:
312:
65537 is the largest known prime number of the form
970:65537 is commonly used as a public exponent in the
1024:
950:
907:
821:
736:
670:
601:
532:
463:
370:
344:
176:
118:(sixty-five thousand five hundred thirty-seventh)
1155:
750:found that the next Fermat number is composite:
974:cryptosystem. Because it is the Fermat number
106:sixty-five thousand five hundred thirty-seven
737:{\displaystyle 2^{2^{4}}+1=2^{16}+1=65537.}
998:65537 is also used as the modulus in some
921:, and currently the largest known integer
401:. The only known prime Fermat numbers are
1022:
671:{\displaystyle 2^{2^{3}}+1=2^{8}+1=257,}
389:with compass and unmarked straightedge.
280:
1122:
602:{\displaystyle 2^{2^{2}}+1=2^{4}+1=17,}
1156:
1133:Sinclair ZX Spectrum Basic Programming
533:{\displaystyle 2^{2^{1}}+1=2^{2}+1=5,}
464:{\displaystyle 2^{2^{0}}+1=2^{1}+1=3,}
1031:. New York: Springer-Verlag. p.
1057:"Sequences by difficulty of search"
13:
1023:Conway, J. H.; Guy, R. K. (1996).
161:
14:
1175:
1084:. OpenSSL Project. Archived from
307:
397:, named after the mathematician
1000:Lehmer random number generators
965:
1116:
1102:
1074:
1049:
1016:
1:
1009:
1128:"Chapter 11. Random numbers"
991:, the common shorthand is "F
7:
1110:"RSA with small exponents?"
345:{\displaystyle 2^{2^{n}}+1}
10:
1180:
1002:, such as the one used by
951:{\displaystyle 10^{n}+27}
266:
253:
240:
227:
214:
201:
188:
144:
134:
122:
110:
100:
42:
21:
383:polygon with 65537 sides
919:JacobsthalâLucas number
917:65537 is also the 17th
952:
909:
823:
738:
672:
603:
534:
465:
372:
346:
294:
178:
953:
925:for which the number
910:
824:
739:
673:
604:
535:
466:
373:
347:
300:is the integer after
291:constructible polygon
284:
179:
929:
848:
756:
683:
614:
545:
476:
407:
391:Johann Gustav Hermes
356:
316:
153:
1027:The Book of Numbers
371:{\displaystyle n=4}
948:
905:
819:
734:
668:
599:
530:
461:
368:
342:
304:and before 65538.
295:
285:Construction of a
174:
287:regular 65537-gon
279:
278:
207:10000000000000001
171:
38:
37:
1171:
1148:
1147:
1142:
1141:
1120:
1114:
1113:
1106:
1100:
1099:
1094:
1093:
1078:
1072:
1071:
1069:
1068:
1059:. Archived from
1053:
1047:
1046:
1030:
1020:
990:
983:
957:
955:
954:
949:
941:
940:
914:
912:
911:
906:
886:
885:
867:
866:
865:
864:
841:
828:
826:
825:
820:
794:
793:
775:
774:
773:
772:
743:
741:
740:
735:
721:
720:
702:
701:
700:
699:
677:
675:
674:
669:
652:
651:
633:
632:
631:
630:
608:
606:
605:
600:
583:
582:
564:
563:
562:
561:
539:
537:
536:
531:
514:
513:
495:
494:
493:
492:
470:
468:
467:
462:
445:
444:
426:
425:
424:
423:
399:Pierre de Fermat
378:). Therefore, a
377:
375:
374:
369:
351:
349:
348:
343:
335:
334:
333:
332:
196:
183:
181:
180:
175:
173:
172:
170:
165:
164:
158:
23:
22:
19:
18:
1179:
1178:
1174:
1173:
1172:
1170:
1169:
1168:
1154:
1153:
1152:
1151:
1139:
1137:
1121:
1117:
1108:
1107:
1103:
1091:
1089:
1080:
1079:
1075:
1066:
1064:
1055:
1054:
1050:
1043:
1021:
1017:
1012:
994:
985:
981:
975:
968:
936:
932:
930:
927:
926:
881:
877:
860:
856:
855:
851:
849:
846:
845:
835:
789:
785:
768:
764:
763:
759:
757:
754:
753:
716:
712:
695:
691:
690:
686:
684:
681:
680:
647:
643:
626:
622:
621:
617:
615:
612:
611:
578:
574:
557:
553:
552:
548:
546:
543:
542:
509:
505:
488:
484:
483:
479:
477:
474:
473:
440:
436:
419:
415:
414:
410:
408:
405:
404:
357:
354:
353:
328:
324:
323:
319:
317:
314:
313:
310:
275:
262:
249:
236:
223:
210:
194:
166:
160:
159:
157:
156:
154:
151:
150:
117:
96:
58:
57:
48:List of numbers
17:
12:
11:
5:
1177:
1167:
1166:
1150:
1149:
1124:Vickers, Steve
1115:
1101:
1073:
1048:
1041:
1014:
1013:
1011:
1008:
992:
977:
967:
964:
960:probable prime
947:
944:
939:
935:
904:
903:67280421310721
901:
898:
895:
892:
889:
884:
880:
876:
873:
870:
863:
859:
854:
833:Fortuné Landry
818:
815:
812:
809:
806:
803:
800:
797:
792:
788:
784:
781:
778:
771:
767:
762:
748:Leonhard Euler
733:
730:
727:
724:
719:
715:
711:
708:
705:
698:
694:
689:
667:
664:
661:
658:
655:
650:
646:
642:
639:
636:
629:
625:
620:
598:
595:
592:
589:
586:
581:
577:
573:
570:
567:
560:
556:
551:
529:
526:
523:
520:
517:
512:
508:
504:
501:
498:
491:
487:
482:
460:
457:
454:
451:
448:
443:
439:
435:
432:
429:
422:
418:
413:
367:
364:
361:
341:
338:
331:
327:
322:
309:
308:In mathematics
306:
277:
276:
273:
270:
264:
263:
260:
257:
251:
250:
247:
244:
238:
237:
234:
231:
225:
224:
221:
218:
212:
211:
208:
205:
199:
198:
192:
186:
185:
169:
163:
148:
142:
141:
138:
132:
131:
126:
120:
119:
114:
108:
107:
104:
98:
97:
59:
56:
55:
50:
44:
43:
40:
39:
36:
35:
32:
29:
27:← 65536
16:Natural number
15:
9:
6:
4:
3:
2:
1176:
1165:
1162:
1161:
1159:
1146:
1135:
1134:
1129:
1125:
1119:
1111:
1105:
1098:
1088:on 2017-03-13
1087:
1083:
1077:
1063:on 2014-07-14
1062:
1058:
1052:
1044:
1042:0-387-97993-X
1038:
1034:
1029:
1028:
1019:
1015:
1007:
1005:
1001:
996:
988:
980:
973:
963:
961:
945:
942:
937:
933:
924:
920:
915:
902:
899:
896:
893:
890:
887:
882:
878:
874:
871:
868:
861:
857:
852:
843:
839:
834:
829:
816:
813:
810:
807:
804:
801:
798:
795:
790:
786:
782:
779:
776:
769:
765:
760:
751:
749:
744:
731:
728:
725:
722:
717:
713:
709:
706:
703:
696:
692:
687:
678:
665:
662:
659:
656:
653:
648:
644:
640:
637:
634:
627:
623:
618:
609:
596:
593:
590:
587:
584:
579:
575:
571:
568:
565:
558:
554:
549:
540:
527:
524:
521:
518:
515:
510:
506:
502:
499:
496:
489:
485:
480:
471:
458:
455:
452:
449:
446:
441:
437:
433:
430:
427:
420:
416:
411:
402:
400:
396:
395:Fermat primes
392:
388:
387:constructible
384:
381:
365:
362:
359:
339:
336:
329:
325:
320:
305:
303:
299:
292:
288:
283:
271:
269:
265:
258:
256:
252:
245:
243:
239:
232:
230:
226:
219:
217:
213:
206:
204:
200:
193:
191:
190:Roman numeral
187:
167:
149:
147:
146:Greek numeral
143:
139:
137:
133:
130:
127:
125:
124:Factorization
121:
115:
113:
109:
105:
103:
99:
95:
92:
89:
86:
83:
80:
77:
74:
71:
68:
65:
62:
54:
51:
49:
46:
45:
41:
34:65538 →
33:
30:
28:
25:
24:
20:
1144:
1138:. Retrieved
1132:
1118:
1104:
1096:
1090:. Retrieved
1086:the original
1076:
1065:. Retrieved
1061:the original
1051:
1026:
1018:
997:
986:
978:
969:
966:Applications
922:
916:
844:
842:showed that
830:
752:
745:
679:
610:
541:
472:
403:
311:
297:
296:
1082:"genrsa(1)"
1004:ZX Spectrum
836: [
268:Hexadecimal
220:10022220022
1140:2022-05-26
1092:2017-05-24
1067:2014-06-14
1010:References
805:4294967297
255:Duodecimal
900:×
831:In 1880,
814:×
746:In 1732,
168:ϝ
1164:Integers
1158:Category
1126:(1983).
102:Cardinal
53:Integers
982:= 2 + 1
817:6700417
380:regular
233:1223225
216:Ternary
197:DXXXVII
140:6,543rd
116:65537th
112:Ordinal
1039:
897:274177
732:65537.
289:. See
246:200001
229:Senary
203:Binary
184:͔ΔÏλζŽ
984:with
958:is a
840:]
302:65536
298:65537
272:10001
259:31B15
242:Octal
136:Prime
129:prime
31:65537
1037:ISBN
1033:139
989:= 4
972:RSA
811:641
663:257
385:is
195:LXV
91:90k
88:80k
85:70k
82:60k
79:50k
76:40k
73:30k
70:20k
67:10k
1160::
1143:.
1130:.
1095:.
1035:.
962:.
946:27
934:10
883:64
838:fr
791:32
718:16
594:17
274:16
261:12
1112:.
1070:.
1045:.
993:4
987:n
979:n
976:F
943:+
938:n
923:n
894:=
891:1
888:+
879:2
875:=
872:1
869:+
862:6
858:2
853:2
808:=
802:=
799:1
796:+
787:2
783:=
780:1
777:+
770:5
766:2
761:2
729:=
726:1
723:+
714:2
710:=
707:1
704:+
697:4
693:2
688:2
666:,
660:=
657:1
654:+
649:8
645:2
641:=
638:1
635:+
628:3
624:2
619:2
597:,
591:=
588:1
585:+
580:4
576:2
572:=
569:1
566:+
559:2
555:2
550:2
528:,
525:5
522:=
519:1
516:+
511:2
507:2
503:=
500:1
497:+
490:1
486:2
481:2
459:,
456:3
453:=
450:1
447:+
442:1
438:2
434:=
431:1
428:+
421:0
417:2
412:2
366:4
363:=
360:n
352:(
340:1
337:+
330:n
326:2
321:2
293:.
248:8
235:6
222:3
209:2
162:M
94:â
64:0
61:â
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
