本帖最后由 蔡家雄 于 2022-6-10 21:00 编辑
【再生差2n素数对 有 无限多组】
设 n, k 均为 固定正整数,且 n 与 k 互素,
设 p1 < p2,且 p1, p2 是 差2n素数对,
使 (p1+n)*k -n =p3 与 (p1+n)*k+n =p4 也是 差2n素数对。
及 (p1+n)*k^2 -n =p5 与 (p1+n)*k^2+n =p6 也是 差2n素数对。
例 n=2, k=9 时的三对 再生差4素数对 有 无限多组,
(11317, 11321) ,(101869, 101873) ,(916837, 916841)
(11863, 11867) ,(106783, 106787) ,(961063, 961067)
(17029, 17033) ,(153277, 153281) ,(1379509, 1379513)
(31267, 31271) ,(281419, 281423) ,(2532787, 2532791)
(108799, 108803) ,(979207, 979211) ,(8812879, 8812883)
(234463, 234467) ,(2110183, 2110187) ,(18991663, 18991667)
(283813, 283817) ,(2554333, 2554337) ,(22989013, 22989017)
(790879, 790883) ,(7117927, 7117931) ,(64061359, 64061363)
(880423, 880427) ,(7923823, 7923827) ,(71314423, 71314427)
【再生等差30的四生素数对 有 无限多组】
设 k 为 固定正整数,且 15 与 k 互素,
设 (p, p+30, p+60, p+90) 是 等差30的四生素数对,
使 (p+45)*k -45, (p+45)*k -15, (p+45)*k+15, (p+45)*k+45 也是 等差30的四生素数对。
例 k=4 时的两对 再生等差30的四生素数对 有 无限多组,
(397429, 397459, 397489, 397519) 与 (1589851, 1589881, 1589911, 1589941)
(2219123, 2219153, 2219183, 2219213) 与 (8876627, 8876657, 8876687, 8876717)
(3686561, 3686591, 3686621, 3686651) 与 (14746379, 14746409, 14746439, 14746469)
(4076951, 4076981, 4077011, 4077041) 与 (16307939, 16307969, 16307999, 16308029)
(4661717, 4661747, 4661777, 4661807) 与 (18647003, 18647033, 18647063, 18647093)
(4968149, 4968179, 4968209, 4968239) 与 (19872731, 19872761, 19872791, 19872821)
(5842841, 5842871, 5842901, 5842931) 与 (23371499, 23371529, 23371559, 23371589)
(7043173, 7043203, 7043233, 7043263) 与 (28172827, 28172857, 28172887, 28172917)
(8682209, 8682239, 8682269, 8682299) 与 (34728971, 34729001, 34729031, 34729061)
例 k=7 时的两对 再生等差30的四生素数对 有 无限多组,
(23, 53, 83, 113) 与 (431, 461, 491, 521)
(41, 71, 101, 131) 与 (557, 587, 617, 647)
(137, 167, 197, 227) 与 (1229, 1259, 1289, 1319)
(12011, 12041, 12071, 12101) 与 (84347, 84377, 84407, 84437)
(15383, 15413, 15443, 15473) 与 (107951, 107981, 108011, 108041)
(74843, 74873, 74903, 74933) 与 (524171, 524201, 524231, 524261)
(98807, 98837, 98867, 98897) 与 (691919, 691949, 691979, 692009)
(141619, 141649, 141679, 141709) 与 (991603, 991633, 991663, 991693)
(184181, 184211, 184241, 184271) 与 (1289537, 1289567, 1289597, 1289627)
(464923, 464953, 464983, 465013) 与 (3254731, 3254761, 3254791, 3254821)
(624007, 624037, 624067, 624097) 与 (4368319, 4368349, 4368379, 4368409)
(891617, 891647, 891677, 891707)与 (6241589, 6241619, 6241649, 6241679)
(1135861, 1135891, 1135921, 1135951) 与 (7951297, 7951327, 7951357, 7951387)
(1140281, 1140311, 1140341, 1140371) 与 (7982237, 7982267, 7982297, 7982327)
(2848663, 2848693, 2848723, 2848753) 与 (19940911, 19940941, 19940971, 19941001)
(4499863, 4499893, 4499923, 4499953) 与 (31499311, 31499341, 31499371, 31499401)
(6637591, 6637621, 6637651, 6637681) 与 (46463407, 46463437, 46463467, 46463497)
(8040601, 8040631, 8040661, 8040691) 与 (56284477, 56284507, 56284537, 56284567)
(9140429, 9140459, 9140489, 9140519) 与 (63983273, 63983303, 63983333, 63983363)
【再生等差2310的六生素数对 有 无限多组】
设 k 为 固定正整数,且 1155 与 k 互素,
设 (p, p+2310, p+4620, p+6930, p+9240, p+11550) 是 等差2310的六生素数对,
使 (p+5775)*k -5775, (p+5775)*k -3465, (p+5775)*k -1155, (p+5775)*k+1155, (p+5775)*k+3465, (p+5775)*k+5775 也是 等差2310的六生素数对。
例 k=13 时的两对 再生等差2310的六生素数对 有 无限多组,
有 (267857, 270167, 272477, 274787, 277097, 279407)
与 (3551441, 3553751, 3556061, 3558371, 3560681, 3562991)
有 (2517227, 2519537, 2521847, 2524157, 2526467, 2528777)
与 (32793251, 32795561, 32797871, 32800181, 32802491, 32804801)
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