## 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.