we list binary irreducible and primitive polynomials, with each polynomial having exactly 3 terms: x^n + x^k + 1 and computations being done in the field GF(2) (the binary field).
the list is a similar to Table 1 of "On Primitive Trinomials (Mod 2)", by Neal Zierler and John Brillhart (1968). in that paper, "Key to Table 1" says primitive k are underlined, but they are actually italicized in the published article. it is difficult to distinguish upright and italic for some digits. presumably they were originally underlined in the manuscript, but the publisher idiotically converted underlining to italics. the entry for n=18, k=9 is an error: the "9" appears italicized, but x^18 + x^9 + 1 is not primitive.
in our table below, primitive k are parenthesized. for example, the entry for 18 is "3 (7) 9". this means x^18 + x^3 + 1 is irreducible but not primitive, x^18 + x^7 + 1 is irreducible and primitive, and x^18 + x^9 + 1 is irreducible but not primitive. the results also hold for n-k by symmetry, so x^18 + x^15 + 1 is irreducible but not primitive, x^18 + x^11 + 1 is irreducible and primitive, and 18 - 9 = 9, so there is nothing new for k=9.
similar computation by others:
https://www.jjj.de/mathdata/all-trinomial-irredpoly-short.txt
https://www.jjj.de/mathdata/all-trinomial-primpoly-short.txt
by default (and the default is not easy to change), the GNU libc (glibc) rand and random functions use the primitive trinomial x^31 + x^3 + 1, doing arithmetic in base 2^32 instead of binary, and "chucking the least random bit". the period is therefore at least 2^31-1 (the comments suggest at least 31*(2^31-1) ), which is not great for 31*32 = 992 bits of state, but at least it's fast. (future work: does arithmetic mod 2^64 work just as fast?)
pseudorandom number generators (PRNG) that do LFSR made wider are called lagged Fibonacci generators. the glibc PRNG above is a lagged Fibonacci, though its documentation does not explicitly say so.
here is C++ code to determine whether a trinomial is primitive by brute force, running a linear feedback shift register through its entire period. much more efficient is the Pari/GP code below.
we first to load up (using addprimes) all known factorizations of base 2 from the Cunningham project so that factorint runs quickly. using Cunningham project data, we also generate a bitvector "unfactored" that indicates that the complete factorization of 2^n-1 is not known. those entries are marked UNFACTORED below. for those entries, using known factors, we can eliminate some irreducible polynomials as not primitive, but those that remain, possibly primitive, are marked with a question mark. the first UNFACTORED and first question mark is at order 1207. the first trinomial definitely not primitive but UNFACTORED is at order 1265.
we first run cunninghamquiet on each entry to compute and cache largest prime factors because they are not explicitly given in the Cunningham tables. we need to factor 2^n+1 for half size and Aurifeuillian 2^n+1 for quarter size to get all the possible prime factors. fortunately, the unfactored "holes" in the latter tables do not affect 2^n-1 below 1500.
not sure why initializing g=Mod(x,p) is correct, but that replicates what Zierler and Brillhart did. in the language of the Fermat primality test, x would be a witness to compositeness.
Pari/GP code:
primesfromcunningham=[...];
addprimes(primesfromcunningham);
unfactored=[...];
factorandcache(n) = my(ff=factorint(n)); my(nf=matsize(ff)[1]); for(j=1, nf, if(ff[j,1]>10^6, addprimes(ff[j,1]))); ff
cunninghamquiet(n) = my(m,k,yL,yM,a); gettime; if(n%2==0, m=n/2; if(m%4==2, k=(m-2)/4; yL=2^(2*k+1)-2^(k+1)+1; yM=2^(2*k+1)+2^(k+1)+1; if(yL*yM != 2^m+1, error("Aurifeuillean failed")); factorandcache(yL); factorandcache(yM)); factorandcache(2^m+1)); factorandcache(2^n-1)
unfactoredtrinomial(n) = print1(" UNFACTORED"); my(cy=2^n-1); my(ff=factorint(cy,1+2+8)); my(nf=matsize(ff)[1]); for(k=1,n/2, my(p=Mod(1,2)*x^n+x^k+1); if(polisirreducible(p), g=Mod(x,p); good=1; for(j=1, nf, eo=cy/ff[j,1]; raised=g^eo; if(raised==1, good=0; break)); if(good, print1(" (",k,"?)"), print1(" ",k))))
regulartrinomial(n) = cunninghamquiet(n); my(cy=2^n-1); my(ff=factorint(cy)); my(nf=matsize(ff)[1]); for(k=1,n/2, my(p=Mod(1,2)*x^n+x^k+1); if(polisirreducible(p), g=Mod(x,p); good=1; for(j=1, nf, eo=cy/ff[j,1]; raised=g^eo; if(raised==1, good=0; break)); if(good, print1(" (",k,")"), print1(" ",k))))
dotrinomial(n) = print1("dataline "n":"); if(unfactored[n], unfactoredtrinomial(n), regulartrinomial(n)); print("")
for(n=2, 1500, dotrinomial(n))
2: (1)
3: (1)
4: (1)
5: (2)
6: (1) 3
7: (1) (3)
8:
9: 1 (4)
10: (3)
11: (2)
12: 3 5
13:
14: 5
15: (1) (4) (7)
16:
17: (3) (5) (6)
18: 3 (7) 9
19:
20: (3) 5
21: (2) 7
22: (1)
23: (5) (9)
24:
25: (3) (7)
26:
27:
28: 1 (3) (9) (13)
29: (2)
30: 1 9
31: (3) (6) (7) (13)
32:
33: 10 (13)
34: 7
35: (2)
36: 9 (11) 15
37:
38:
39: (4) (8) (14)
40:
41: (3) (20)
42: 7
43:
44: 5
45:
46: 1
47: (5) (14) (20) (21)
48:
49: (9) (12) (15) (22)
50:
51:
52: (3) 7 (19) (21)
53:
54: 9 21 27
55: 7 (24)
56:
57: 4 (7) (22) 25
58: (19)
59:
60: (1) 9 (11) 15 17 23
61:
62: 29
63: (1) (5) 11 28 (31)
64:
65: (18) (32)
66: 3
67:
68: (9) (33)
69:
70:
71: (6) (9) (18) (20) (35)
72:
73: (25) (28) (31)
74: 35
75:
76: 21
77:
78:
79: (9) (19)
80:
81: (4) (16) (35)
82:
83:
84: 5 9 11 (13) 27 35 39
85:
86: 21
87: (13)
88:
89: (38)
90: 27
91:
92: 21
93: (2)
94: (21)
95: (11) (17)
96:
97: (6) (12) (33) (34)
98: (11) (27)
99:
100: 15 19 25 (37) 49
101:
102: 29 37
103: (9) (13) (30) (31)
104:
105: 4 7 8 (16) (17) 28 (37) (43) 49 (52)
106: (15)
107:
108: 17 27 (31) 33 45
109:
110: 33
111: (10) (49)
112:
113: (9) (15) (30)
114:
115:
116:
117:
118: (33) (45)
119: (8) (38)
120:
121: (18) 30
122:
123: (2)
124: 19 (37) 45 55
125:
126: 21 49
127: (1) (7) (15) (30) (63)
128:
129: (5) (31) (46)
130: (3)
131:
132: 17 (29)
133:
134: (57)
135: (11) (16) (22) 29
136:
137: (21) (35) (57)
138:
139:
140: 15 (29) 45 65
141:
142: (21)
143:
144:
145: (52) (69)
146: 71
147: 14 49
148: (27)
149:
150: (53) 73
151: (3) (9) (15) (31) (39) (43) (46) (51) (63) (66) (67) (70)
152:
153: (1) (8)
154: 15
155: 62
156: 9 11 57 61 63 65
157:
158:
159: (31) (34) (40)
160:
161: (18) (39) (60)
162: 27 63 81
163:
164:
165:
166: 37
167: (6) (35) (59) (77)
168:
169: (34) (42) (57) (84)
170: 11 (23)
171:
172: 1 (7) 81
173:
174: (13) 57
175: (6) (16) (18) (57)
176:
177: (8) (22) (88)
178: 31 (87)
179:
180: 3 27 33 45 55 69
181:
182: 81
183: (56)
184:
185: (24) (41) (69)
186: 11 79
187:
188:
189:
190:
191: (9) (18) (51) (71)
192:
193: (15) (73) (85)
194: (87)
195:
196: 3 33 67
197:
198: 9 (65)
199: (34) (67)
200:
201: (14) (17) (59) (79)
202: (55)
203:
204: 27 99
205:
206:
207: (43)
208:
209: (6) (8) (14) (45) (47) (50) (62)
210: 7
211:
212: (105)
213:
214: 73
215: (23) (51) (63) (77) (101)
216:
217: (45) (64) (66) (82) (85)
218: (11) (15) (71) (83) 99
219:
220: 7 33 49
221:
222:
223: (33) (34) (64) (70) (91)
224:
225: (32) (74) (88) (97) (109)
226:
227:
228: 113
229:
230:
231: (26) (34) 91
232:
233: (74)
234: (31) (103)
235:
236: (5)
237:
238: 73 117
239: (36) (81)
240:
241: (70)
242: 95
243:
244: 111
245:
246:
247: (82) (102)
248:
249: 35 (86)
250: (103)
251:
252: 15 27 33 39 53 59 (67) 77 81 105 109 117
253: 46
254:
255: (52) (56) (82)
256:
257: (12) (41) (48) (51) (65)
258: 71 (83)
259:
260: 15 35 95 105
261:
262:
263: (93)
264:
265: (42) (127)
266: (47)
267:
268: (25) (61)
269:
270: (53) 81 (133)
271: (58) (70)
272:
273: (23) 28 (53) 55 56 (67) (88) (92) 98 (110) (113)
274: (67) (99) (135)
275:
276: 63 91
277:
278: (5)
279: (5) (10) (38) (40) (41) (59) 73 (76) (80) (125)
280:
281: (93) (99)
282: (35) (43) 63
283:
284: 53 99 (119) 141
285:
286: (69) (73)
287: (71) (116) (125)
288:
289: (21) (36) (84)
290:
291:
292: 37 (97)
293:
294: 33 49 (61) 81
295: (48) (112) (123) (142) (147)
296:
297: (5) (83) (103) (122) (137)
298:
299:
300: 5 (7) 45 55 57 (73) 75 (91) 111 147
301:
302: (41)
303: 1
304:
305: (102)
306:
307:
308: 15
309:
310: 93
311:
312:
313: (79) (121)
314: (15)
315:
316: 63 (135)
317:
318: 45
319: (36) (52) (129)
320:
321: (31) 41 (56) (76) (82) (155)
322: (67)
323:
324: 51 81 93 99 135 149
325:
326:
327: (34) (152)
328:
329: (50) (54)
330: 99
331:
332: 89 (123)
333: (2)
334:
335:
336:
337: (55) (57) (135) (139) (147)
338:
339:
340: 45 165
341:
342: (125) 133
343: (75) (135) (138) (159)
344:
345: (22) (37) (106)
346: 63
347:
348: 103
349:
350: (53)
351: (34) (55) 79 (116) (134)
352:
353: (69) (95) (138) (143) (153) (173)
354: 99 135
355:
356:
357:
358: 57
359: (68) (117)
360:
361:
362: (63) (107)
363:
364: 9 (67)
365:
366: (29)
367: (21) (171)
368:
369: (91) (110)
370: (139) (183)
371:
372: 111 135 165
373:
374:
375: (16) (64) (149) (182)
376:
377: (41) (75)
378: (43) 63 (107) 147
379:
380: (47) 63 105
381:
382: (81)
383: (90) (108) (135)
384:
385: (6) (24) (51) (54) 111 (142) (159)
386: (83)
387:
388: 159
389:
390: 9 49 (89)
391: (28) (31)
392:
393: (7) (62) (91)
394: (135)
395:
396: (25) 51 87 (109) (169) (175)
397:
398:
399: 26 49 (86) (109) 154 (181)
400:
401: (152) (170)
402: 171
403:
404: 65 (189)
405:
406: 141 (157)
407: (71) (105)
408:
409: (87)
410:
411:
412: (147)
413:
414: 13 53
415: (102) (163)
416:
417: (107) (113) (155)
418: 199
419:
420: 7 45 65 77 87 127 135 161 195
421:
422: (149) 177
423: (25)
424:
425: (12) (21) (42) (66) (111) (191)
426: 63
427:
428: (105)
429:
430:
431: (120) (200)
432:
433: (33) (61) (118) (153)
434:
435:
436: (165)
437:
438: (65)
439: (49) (133) (145) (156) (171)
440:
441: 7 (31) 35 (127) 196 (212) 217
442:
443:
444: 81
445:
446: (105) (153)
447: (73) (83)
448:
449: (134) (167)
450: 47 (79) 159
451:
452:
453:
454:
455: (38) (62) (74)
456:
457: (16) (61) (123) (210) (217) (226)
458: (203)
459:
460: 19 (61)
461:
462: (73)
463: (93) (168) (214)
464:
465: 31 (59) (103) 124 (158) 217
466:
467:
468: 27 33 143 171 183 189 195
469:
470: 9 (149) (177)
471: (1) (119) (127)
472:
473: 200
474: (191) (215)
475:
476: 9 (15) 129 (141)
477:
478: (121)
479: (104) (105) (122) (158) (224)
480:
481: (138) (201) (231)
482:
483:
484: (105)
485:
486: 81 189 243
487: (94) (127)
488:
489: (83)
490: (219)
491:
492: 7
493:
494: 17 (137)
495: (76) (89) (118) (226)
496:
497: (78) (216) (228)
498: 155
499:
500: 27 75 95 125 185 243 245
501:
502:
503: (3) (26) (248)
504:
505: (156) (174)
506: 23 (95) (135)
507:
508: 9 (109)
509:
510: 69 197
511: (10) (15) (31) (160) (202) (216)
512:
513: 26 (85) (175) 242
514: 67 103
515:
516: 21 91
517:
518: (33) (45) 113
519: (79)
520:
521: (32) (48) (158) (168)
522: 39 171 259
523:
524: (167) 195
525:
526: 97
527: (47) (123) (147) (152) (198) (239)
528:
529: (42) (114) (157)
530:
531:
532: (1) (37) 81
533:
534: 161 261
535:
536:
537: (94)
538: 195
539:
540: 9 11 81 99 113 135 155 165 (179) 191 207 (211)
541:
542:
543: (16) (28) (58) (203) (235)
544:
545: (122)
546:
547:
548:
549:
550: (193)
551: (135) (240)
552:
553: (39) (57) (94) (99) (109) (255) (258)
554:
555:
556: (153) 273
557:
558: 73
559: (34) (70) (148) (210)
560:
561: (71) (109) (155)
562:
563:
564: (163)
565:
566: (153) 273
567: 28 112 (143) 245 (275)
568:
569: (77) (210)
570: (67) 143
571:
572:
573:
574: (13)
575: (146) 258
576:
577: (25) (27) (231)
578:
579:
580: 237
581:
582: (85) 261
583: (130)
584:
585: 88 (121) (151) (157) (232) 256
586:
587:
588: 35 77 91 99 (151) 201 245 (253)
589:
590: (93)
591:
592:
593: (86) (108) (119) (177)
594: (19) 27 (35) 195
595:
596: 273
597:
598:
599: (30) (210)
600:
601: (201) (202)
602: 215
603:
604: 105
605:
606: 165
607: (105) (147) (273)
608:
609: (31) 91 (128) (181) (233)
610: (127)
611:
612: 81 157 297
613:
614: 45 177
615: (211) (232) (238)
616:
617: (200)
618: 295
619:
620: 9 93 95 185
621:
622: (297)
623: (68) (87) (128) (185) (230) (251) (296) (311)
624:
625: (133) (156)
626: 251
627:
628: (223) (289)
629:
630:
631: (307)
632:
633: (101) (292)
634: 39 (315)
635:
636: 217 269 311 315
637:
638:
639: (16) (88) (95) (179) 224 233 295 (305)
640:
641: (11) (36) (45) (95) (287)
642: (119)
643:
644:
645:
646: (249)
647: (5) (150) (215) (312)
648:
649: (37) (73) (171) (310) (321)
650: (3)
651: 14
652: (93) (97)
653:
654: 33 45 213 249
655: (88) (192)
656:
657: (38) 73 (92) (148) 292
658: (55) 163
659:
660: 11 21 99 121 145 253
661:
662: 21 141 269 (297)
663: 107 (257) 275 (307)
664:
665: (33) (53) (144) (192) (269) (317)
666:
667:
668: 147
669:
670: (153) (273)
671: (15) (201) (243)
672:
673: (28) (183) (252) (259) (300)
674:
675:
676: 31 (241) (277)
677:
678:
679: (66) (216)
680:
681:
682: 171 243
683:
684: 209
685:
686: (197)
687: (13) (133) 274
688:
689: (14) (87) (179) (207) (336)
690: 79
691:
692: (299) 311
693:
694: 169
695: 177 (212)
696:
697: (267) (310)
698: (215) (311)
699:
700: 75 145 225 325
701:
702: (37) 93 309 (317)
703:
704:
705: 17 (19) 68 79 (161) (194) (266) (328) (331)
706:
707:
708: 15 (287) (301) 335
709:
710:
711: (92) 319
712:
713: (41) (297)
714: (23) (151) 203 259
715:
716: (183) 257 (275)
717:
718: 165
719: (150) (174) (257) (299) (314)
720:
721: (9) (159) (256) (270) (283) (328)
722: (231)
723:
724: 207
725:
726: (5) (241)
727: (180) (217) (357)
728:
729: (58) (253)
730: (147)
731:
732: 343
733:
734:
735: (44) 49 (89) 112 119 196 259 (262) 299 301 343 364
736:
737: (5) (303)
738: (347)
739:
740: 135 (153) 287 (317)
741:
742: 85 225
743: (90) (144) (146) (209) (210) (239) (279) (326)
744:
745: (258) 333 (336) (342)
746: (351)
747:
748: 19 31 133
749:
750: 309
751: (18) (187) (310)
752:
753: (158)
754: (19) (147)
755:
756: 45 81 99 117 119 159 201 217 231 243 315 (349) 351
757:
758: 233 357
759: (98) (109) 230 (251) 299
760:
761: (3) (33) (84) (138)
762: (83) 115
763:
764:
765:
766:
767: (168) (254)
768:
769: (120) (216) (322)
770:
771:
772: (7) (121)
773:
774: (185) 249
775: 93 217 (367)
776:
777: (29) 70 (302) 343
778: (375)
779:
780: 13 45 143 221 285 299 301 305 315
781:
782: (329)
783: (68) (71) (103) 202
784:
785: (92) (191) (212) (219)
786:
787:
788:
789:
790:
791: (30) (108) (251)
792:
793: (253)
794: (143) 263
795:
796:
797:
798: 53 141
799: (25)
800:
801: (217) (325)
802:
803:
804: 75 183 (295)
805:
806: 21 (141)
807: (7) (308) 403
808:
809: (15) (92) (137) (210) (233)
810: 159 243 247 (299) 399
811:
812: 29 87 125 (167) (183) (237) 261 377
813:
814: 21 (145)
815: (333) (336) (339)
816:
817: (52) (187) (355)
818: (119)
819:
820: 123
821:
822: 17 201 297 405
823: (9) (91) (280) (357)
824:
825: (38)
826: (255)
827:
828: 189 (205)
829:
830:
831: (49) (322)
832:
833: (149) (159) (195) (215) 339
834: 15
835:
836:
837:
838: (61)
839: (54) (314) (327)
840:
841: (144) (309)
842: (47)
843:
844: 105
845: (2)
846: 105 129 189
847: (136) (153) (276) (393)
848:
849: (253)
850: (111) (139)
851:
852: 159 297 357
853:
854:
855: (29) (142) (146) (151) (254) 377 (379)
856:
857: (119) (215) (221) (258) (270) (402)
858: 207 219
859:
860: 35 405
861: 14
862: (349)
863:
864:
865: (1) (9) (228) (379) (417)
866: (75) 215
867:
868: (145) 241 285
869:
870: 301
871: (378)
872:
873: 352
874:
875:
876: 149 291
877:
878:
879: (11) (80) (91) (121) (190) 238
880:
881: (78) (84) (236) (392)
882: 99 147 183 243 343
883:
884: (173)
885:
886:
887: (147) (317) (336)
888:
889: 127 (169) (310) (312) 381
890: 183
891:
892: (31)
893:
894: (173) 413
895: (12)
896:
897: (113) (382)
898: (207)
899:
900: (1) 15 21 79 135 165 171 219 225 233 273 275 333 441
901:
902: 21
903: 35 (160) 217 (220) (221) (263) (278) 322
904:
905: (117) (341)
906: 123 (187)
907:
908: (143)
909:
910:
911: (204) (260) (378)
912:
913: (91) (129) (439)
914:
915:
916: 183 243
917:
918: (77)
919: (36) (141) (274) (336) (390)
920:
921: (221) (340) (364) (455)
922:
923:
924: 31 45 173 203 215
925:
926: (365)
927: (403) 433 (455)
928:
929:
930: 31 279
931:
932: 177 231 (275) (387) 447 (455)
933:
934:
935: (417)
936:
937: (217) (316)
938: (207)
939:
940:
941:
942: 45
943: (24) (334) 450
944:
945: 77 (79) (94) 112 154 187 (191) 203 (229)
946:
947:
948: 189 405 461
949:
950:
951: (260) (290) (391)
952:
953: (168) (224) (435)
954: 131 135
955:
956: (305) (347)
957:
958:
959: (143) (164) (312) (395) (422)
960:
961: (18) (130) (393)
962:
963:
964: (103) 231 313 433 (441)
965:
966: 201
967: (36) (130) (210) (400)
968:
969: 31 (74) (308) 404 (446)
970:
971:
972: 7 (115) 153 (155) 243 279 297 377 405
973:
974:
975: (19) 121 133
976:
977: (15) (63) (207) (374) (480)
978:
979: 178
980:
981:
982: 177 (277) 481
983: (230) (342)
984:
985: (222)
986: 3
987:
988: (121)
989:
990: 161 297
991: (39) (171) (193) (466) (472)
992:
993: (62) 367
994: (223)
995:
996: 65 369
997:
998: (101) 417 477
999: (59)
1000:
1001: (17) (54) (354) (422)
1002:
1003:
1004:
1005:
1006:
1007: (75) (96) (351) (386) (405)
1008:
1009: (55) (148)
1010: 99
1011:
1012: 115 301
1013:
1014: (385)
1015: (186) (258) (447) (466) (484)
1016:
1017:
1018:
1019:
1020: 135 (461) 495
1021:
1022: (317)
1023: (7) (43) (127)
1024:
1025: (294) (306) (383)
1026: (35) 375 399
1027:
1028: 119 (203)
1029: 98 343
1030: (93)
1031: (68) (116) (287) (371)
1032:
1033: (108) (330) (340) (498)
1034: (75)
1035:
1036: (411)
1037:
1038:
1039: (21) (88) (279) (364) (418)
1040:
1041: (412)
1042: (439)
1043:
1044: (41) 71 361
1045:
1046:
1047: (10) (430) 470
1048:
1049: (141) (227) (290) (293) (327) (357) (390)
1050: 159 371
1051:
1052: (291) 357
1053:
1054: (105)
1055: (24) (243)
1056:
1057: (198) (331) (438)
1058: (27)
1059:
1060: 439 525
1061:
1062: 49 297 405
1063: (168) (208) (285) (370)
1064:
1065: (463) (476)
1066:
1067:
1068:
1069:
1070:
1071: 7 (50) 56 (281) (436)
1072:
1073:
1074:
1075:
1076:
1077:
1078: 361 (445)
1079: (230) (282) (342)
1080:
1081: (24) 235 (318) (348) 423 (525)
1082: (407)
1083:
1084: (189)
1085: 62
1086: 189 217 321
1087: (112) (240) (445) (457) (481)
1088:
1089: (91) (148) (206) (283) (400) (466)
1090: (79) 243
1091:
1092: (23) 65 77 (113) 143 169 201 353 455 479
1093:
1094: 57 (261)
1095: (139) (383) (476) (538)
1096:
1097: (14) (86) (303)
1098: (83) 87 503
1099:
1100: 35 93 165 209 245 407 497 519
1101:
1102: (117)
1103: (65) (71) (164) (189) (434) (465)
1104:
1105: (21) (57) (96) 321 (363) 439
1106: (195)
1107:
1108: (327)
1109:
1110: 417 549
1111: (13) (40) (238) (321) 531
1112:
1113: (107) 217 238 280 (310)
1114:
1115:
1116: 59 227 333 405 (479) 495
1117:
1118:
1119: (283) (329)
1120:
1121: (62) (107) (113) (176) (203)
1122: 427
1123:
1124:
1125:
1126: 105 (309) 561
1127: (27) (171) (317) (465) (512)
1128:
1129: (103) (135) (208)
1130: (551)
1131:
1132:
1133:
1134: 129 189 321 441
1135: (9) (36) (106) (298)
1136:
1137: (277) (313) 458
1138: (31) (183)
1139:
1140: 141 189 413 419 (539)
1141:
1142: (357) 461
1143:
1144:
1145: (227) (506)
1146: (131) 243
1147:
1148: (23) 221
1149:
1150:
1151: (90) (125)
1152:
1153: (241) (268) (415) (499) (541)
1154: (75)
1155:
1156: (307) 513
1157:
1158: (245) 249
1159: (66) (129) (424)
1160:
1161: (365) (409) (551)
1162:
1163:
1164: (19) 85 161
1165:
1166: (189) (513)
1167: (133) (265)
1168:
1169: (114) (173) (207)
1170: 27 267
1171:
1172:
1173:
1174: (133)
1175: (476)
1176:
1177: (16) (64) (70) (126) (186) (288) (412)
1178: (375) (491)
1179:
1180: 25 99
1181:
1182: 77 405 425
1183: (87) (108) (136)
1184:
1185: (134) (317) (386)
1186: (171)
1187:
1188: 75 153 251 261 287 327 395 (413) 507 525
1189:
1190: (233)
1191: (196) (212)
1192:
1193: (173)
1194:
1195:
1196: 281 (519)
1197:
1198: 405
1199: (114)
1200:
1201: (171) (360) (388)
1202: (287)
1203:
1204: 43 129 387 459 559
1205:
1206: 513
1207: UNFACTORED (273?) (423?) (511?)
1208:
1209: (118) (145) (290) (356)
1210: (243) 363 (463)
1211:
1212: (203) 567
1213: UNFACTORED
1214: (257) (605)
1215: (302) 322
1216:
1217: UNFACTORED (393?)
1218: 91 215 471
1219:
1220: (413) 555
1221:
1222:
1223: (255) (549) (588)
1224:
1225: (234) (547)
1226: (167)
1227:
1228: (27) 193
1229: UNFACTORED
1230: 433
1231: UNFACTORED (105?) (265?) (355?) (390?) (453?) (496?) (517?)
1232:
1233: (151) (172) (247) (256) (368) (520)
1234: (427)
1235:
1236: 49 119 (151) 301 317 441
1237: UNFACTORED
1238: (153)
1239: (4) (10) 56 58 (146) 154 (193) (466) (610) 616
1240:
1241: (54) (165) (501)
1242: 203 (395) (403) 479
1243: UNFACTORED
1244:
1245:
1246: (25) (69) (421)
1247: (14) (27) (90) (585)
1248:
1249: UNFACTORED (187?) (237?)
1250:
1251:
1252: 97 265 409 577
1253: UNFACTORED
1254:
1255: UNFACTORED (589?)
1256:
1257: (289) (347) (595)
1258:
1259: UNFACTORED
1260: 21 135 195 231 261 295 335 381 385 405 421 483 545 585 589
1261:
1262:
1263: (77) (346)
1264:
1265: UNFACTORED (119?) (338?) (371?) 467 552 (576?)
1266: 7 447
1267:
1268: 345
1269:
1270: 333
1271: (17) (53) (380) (450)
1272:
1273: (168) (357) (495)
1274:
1275:
1276: 217 (427) 541
1277: UNFACTORED
1278: 189 (385) (637)
1279: (216) (418)
1280:
1281: 229 392 (547) (596)
1282: (231)
1283: UNFACTORED
1284: (223) 315
1285:
1286: (153) 441
1287: (470) (622)
1288:
1289: (99) (204) (212) (242) (609)
1290:
1291: UNFACTORED
1292:
1293:
1294: 201
1295: (38)
1296:
1297: UNFACTORED (198?) (337?) (565?)
1298: (399)
1299:
1300: 75 175 (217) 475 525 607
1301:
1302: 77 (325)
1303:
1304:
1305: 326 (418)
1306: (39)
1307:
1308: 495 631
1309:
1310: (333)
1311: (476) (569)
1312:
1313: (164) (354) (389)
1314: 19 195 511
1315: UNFACTORED
1316:
1317:
1318:
1319: UNFACTORED (129?)
1320:
1321: (52) (490)
1322:
1323:
1324: (337) (555)
1325: UNFACTORED
1326: 397
1327: (277) (372) (379) (466) (640)
1328:
1329: 73 (364)
1330:
1331: UNFACTORED
1332: (95) 193 243 407
1333:
1334: 617
1335: (392) (647)
1336:
1337: (75) (102) 344 (408)
1338: 315 459 (511)
1339: UNFACTORED
1340: 125 (189) 305 507 539
1341:
1342:
1343: UNFACTORED (348?) (360?)
1344:
1345: UNFACTORED (553?)
1346:
1347:
1348: (553)
1349: UNFACTORED
1350: 237 477 629
1351: UNFACTORED (39?) (150?) (201?) (241?) (382?) (508?) (565?)
1352:
1353: 371 (607)
1354: (255)
1355:
1356: 131 (275)
1357: UNFACTORED
1358: 117
1359: (98) (140) (455) (533) (595)
1360:
1361: (56) (75) (195) (378)
1362: (655)
1363:
1364: 239 285 497
1365:
1366: (1)
1367: UNFACTORED (134?) (567?)
1368:
1369: (88) (360) (427)
1370:
1371:
1372: (181)
1373:
1374: 609
1375: (52) (126) (321) (397) (586) (637) (684)
1376:
1377: (100) (248)
1378:
1379: UNFACTORED
1380: 183 455
1381: UNFACTORED
1382:
1383: (130)
1384:
1385: UNFACTORED (12?) (171?) (237?) (608?) (617?) (659?)
1386: 219 455 599
1387: UNFACTORED
1388: 11 417
1389:
1390: (129) (577)
1391: (3) (84) (236) (363) (525)
1392:
1393: (300) (342) (663)
1394:
1395:
1396: 97 (339)
1397:
1398: 601
1399: (55) (220) (259) (264) (540) (589)
1400:
1401: (92) 364 (550) (679)
1402: (127) (355)
1403: UNFACTORED
1404: 81 99 221 403 429 513 549 567 569 585 (661)
1405:
1406:
1407: (47) 98 119 (260) (272) 413 553 620
1408:
1409: (194) (464)
1410: (383) 447 531
1411:
1412: 125 (153)
1413:
1414: (429) (589) (649)
1415: UNFACTORED (282?) (377?) (392?)
1416:
1417: UNFACTORED (342?) (466?) (682?)
1418:
1419: UNFACTORED
1420: 33 273 595 705
1421: UNFACTORED
1422: 49 497 573 645
1423: UNFACTORED (15?) (228?) (289?) (310?) (334?) (631?)
1424:
1425: 28 (88) 343 (611) (616)
1426: (103)
1427:
1428: 27 45 423 (557) 631
1429: UNFACTORED
1430: 33 621
1431: (17) (332) (445) (514) (655)
1432:
1433: UNFACTORED (387?)
1434: 363 503
1435:
1436: 83
1437:
1438: (357)
1439: UNFACTORED
1440:
1441: (322) (442) (465)
1442: 395
1443: UNFACTORED
1444: 595
1445:
1446: 421
1447: UNFACTORED (195?) (301?) (342?) (663?) (721?)
1448:
1449: (13) (83) (176) 301 (515) (521) (706)
1450:
1451: UNFACTORED
1452: 315
1453: UNFACTORED
1454: (297)
1455: (52) 182 (307)
1456:
1457: (314) (422) (567)
1458: 243 567 729
1459:
1460: 185 485 627
1461:
1462:
1463: UNFACTORED (575?) (645?)
1464:
1465: UNFACTORED (39?) 522
1466: (311)
1467: UNFACTORED
1468: (181) 439 (709)
1469:
1470: 49 101 (569) 649 657 733
1471: (25) (393) (622)
1472:
1473: (77) (143) (271) (311) (488)
1474:
1475: UNFACTORED
1476: 21 (265) 467 665
1477: UNFACTORED
1478: (69)
1479: (49) (50) (280) (368) 434
1480:
1481: UNFACTORED (32?) (183?) (311?) (716?)
1482: 411
1483: UNFACTORED
1484:
1485:
1486: (85)
1487: (140) (237) (276) (717)
1488:
1489: (252)
1490: 279
1491:
1492: 307
1493: UNFACTORED
1494:
1495: (94) (241) (561) (679)
1496:
1497: UNFACTORED (49?) (332?) (448?) (563?) (608?) (685?)
1498:
1499: UNFACTORED
1500: 25 35 225 275 285 365 375 455 555 735 749
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