Saturday, November 30, 2013

Small pairs of primitive roots with sum and difference one - raw data for primes up to 1013



Prime p
Primitive roots
Category 1
Sum 1
Small (least < \/p )
Category 2
Difference 1
Small (least < \/p )
2
1
no
no
3
2
no
no
5
2, 3
no
(2,3)
7
3, 5
no
no
11
2, 6, 7, 8
no
no
13
2, 6, 7, 11
no
no
17
3, 5, 6, 7, 10, 11, 12, 14
no
no
19
2, 3, 10, 13, 14, 15
no
(2,3)
23
5, 7, 10, 11, 14, 15, 17, 19, 20, 21
no
no
29
2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27
(3,27)
(2,3)
31
3, 11, 12, 13, 17, 21, 22, 24
no
no
37
2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, 35
no
no
41
6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35
no
no
43
3, 5, 12, 18, 19, 20, 26, 28, 29, 30, 33, 34
no
no
47
5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39, 40, 41, 43, 44, 45
(5,43)
no
53
2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 33, 34, 35, 39, 41, 45, 48, 50, 51
(3,51)
(2,3)


From now on, the set of all primitive roots (column 2 above) is not listed.



59
(6, 54)
no
61
(7, 55)
(6, 7)
67
(7, 61)
no
71
(7, 65)
no
73
no
no
79
(3, 77)
(6, 7)
83
(5, 79)
(5, 6)
89
(7, 83)
(6, 7)
97
no
no
101
(3, 99)
(2, 3)
103
(5, 99)
(5, 6)
107
(5, 103)
(5, 6)
109
no
(10, 11)
113
(6, 108)
(5, 6)
127
no
(6, 7)
131
(6, 126)
no
137
(6, 132)
(5, 6)
139
no
(2, 3)
149
(3, 147)
(2, 3)
151
(6, 146)
(6, 7)
157
(6, 152)
(5, 6)
163
(11, 153)
(11, 12)
167
(5, 163)
no
173
(3, 171)
(2, 3)
179
(6, 174)
(6, 7)
181
no
no
191
no
no
193
no
no
197
(3, 195)
(2, 3)
199
(3, 197)
no
211
(7, 205)
(2, 3)
223
(10, 214)
(5, 6)
227
(5, 223)
(13, 14)
229
(7, 223)
(6, 7)
233
(6, 228)
(5, 6)





239
(13, 227)
(13, 14)
241
(14, 228)
(13, 14)
251
(14, 238)
no
257
(6, 252)
(5, 6)
263
(5, 259)
(14, 15)
269
(3, 267)
(2, 3)
271
(15, 257)
no
277
(6, 272)
(5, 6)
281
(12, 270)
(11, 12)
283
(12, 272)
no
293
(3, 291)
(2, 3)
307
no
(13, 14)
311
(17, 295)
no
313
(15, 299)
(14, 15)
317
(3, 315)
(2, 3)
331
no
no
337
no
no
347
(17, 331)
(5, 6)
349
no
no
353
(13, 341)
(12, 13)
359
(13, 347)
(13, 14)
367
(19, 349)
(10, 11)
373
(15, 359)
(5, 6)
379
(17, 363)
(2, 3)
383
(13, 371)
(10, 11)
389
(15, 375)
(14, 15)
397
(6, 392)
(5, 6)
401
(13, 389)
(12, 13)
409
no
no
419
(10, 410)
(10, 11)
421
no
no
431
no
(13, 14)
433
(15, 419)
(14, 15)
439
no
no
443
(11, 433)
(5, 6)



449
(13, 437)
(12, 13)
457
no
no
461
(3, 459)
(2, 3)
463
(3, 461)
(19, 20)
467
(5, 463)
(5, 6)
479
(13, 467)
no
487
(3, 485)
(10, 11)
491
(6, 486)
(6, 7)
499
(15, 485)
(10, 11)
503
(5, 499)
(19, 20)
509
(3, 507)
(2, 3)
521
(7, 515)
(6, 7)
523
(5, 519)
(21, 22)
541
(14, 528)
(13, 14)
547
(5, 543)
(17, 18)
557
(3, 555)
(2, 3)
563
(5, 559)
(5, 6)
569
(12, 558)
(11, 12)
571
(10, 562)
(17, 18)
577
no
no
587
(5, 583)
(5, 6)
593
(6, 588)
(5, 6)
599
(7, 593)
(22, 23)
601
no
no
607
(3, 605)
(23, 24)
613
(15, 599)
(13, 14)
617
(13, 605)
(12, 13)
619
no
(10, 11)
631
no
(12, 13)
641
(24, 618)
(23, 24)
643
no
(13, 14)
647
(15, 633)
(10, 11)
653
(3, 651)
(2, 3)
659
(10, 650)
(6, 7)



661
no
no
673
(20, 654)
(19, 20)
677
(3, 675)
(2, 3)
683
(11, 673)
(5, 6)
691
no
(12, 13)
701
(3, 699)
(2, 3)
709
(23, 687)
(22, 23)
719
(22, 698)
(22, 23)
727
no
(19, 20)
733
(24, 710)
(23, 24)
739
(7, 733)
(18, 19)
743
(5, 739)
(13, 14)
751
(3, 749)
(14, 15)
757
(6, 752)
(5, 6)
761
(12, 750)
(6, 7)
769
(14, 756)
(13, 14)
773
(3, 771)
(2, 3)
787
(5, 783)
(12, 13)
797
(3, 795)
(2, 3)
809
(12, 798)
(11, 12)
811
(10, 802)
no
821
(3, 819)
(2, 3)
823
(3, 821)
(22, 23)
827
(13, 815)
(6, 7)
829
(22, 808)
(21, 22)
839
(26, 814)
no
853
(18, 836)
(17, 18)
857
(7, 851)
(5, 6)
859
(18, 842)
(2, 3)
863
(5, 859)
(10, 11)
877
(14, 864)
(13, 14)
881
(24, 858)
(14, 15)
883
(11, 873)
no



887
(5, 883)
(29, 30)
907
(7, 901)
(2, 3)
911
no
no
919
no
(14, 15)
929
(7, 923)
(6, 7)
937
(11, 927)
(10, 11)
941
(3, 939)
(2, 3)
947
(5, 943)
(5, 6)
953
(6, 948)
(5, 6)
967
(5, 963)
(5, 6)
971
(6, 966)
(10, 11)
977
(6, 972)
(12, 13)
983
(5, 979)
(10, 11)
991
(6, 986)
(6, 7)
997
(29, 969)
(28, 29)
1009
no
no
1013
(18, 996)
(17, 18)



CONJECTURE ON SMALL SPECIAL PAIRS OPF PRIMITIVE ROOTS: For all large enough primes p, one can find "small" pairs of primitive roots with sum 1 as well as small pairs of primitive roots with difference 1.