判定梅森质数的卢卡斯序列

cz1

请问Treenewbee先生：
33199^3=2833^3+19081^3+30941^3=15187^3+24197^3+26647^3

不知道等式中的2833^3，19081^3，30941^3，15187^3，24197^3，26647^3，
也可以分为三个立方数的和吗？请教
2833---{450, 2001, 2446}, {744, 2001, 2428}, {1362, 2001, 2302}
19081^3 =a^3+b^3+c^3 可能没有解，
30941^3 =a^3+b^3+c^3 可能没有解，
15187^3 =a^3+b^3+c^3 可能没有解，
24197^3 =a^3+b^3+c^3 可能没有解，
26647^3 =a^3+b^3+c^3 可能没有解，

Treenewbee

yangchuanju 发表于 2023-2-4 06:54
蔡家雄两方程有解，最小解是m=353，非651也！
m^3=a^3+b^3+c^3+d^3
m^4=e^4+f^4+g^4+ h^4

('353^3', 14, [[2, 91, 153, 341], [10, 21, 203, 329], [26, 133, 257, 291], [26, 162, 185, 322], [29, 87, 154, 341], [34, 116, 257, 294], [35, 230, 241, 261], [45, 227, 228, 273], [49, 110, 230, 312], [63, 166, 257, 281], [81, 203, 250, 269], [105, 186, 213, 299], [112, 134, 153, 332], [196, 209, 222, 254]])

('353^4', 1, [[30, 120, 272, 315]])

Treenewbee

('651^2', 15127, [[1, 2, 36, 650], [1, 2, 164, 630], [1, 4, 378, 530], [1, 12, 34, 650], [1, 14, 198, 620], [1, 14, 348, 550], [1, 20, 30, 650], [1, 20, 78, 646], [1, 20, 106, 642], [1, 20, 366, 538], [1, 20, 414, 502], [1, 20, 450, 470], [1, 22, 404, 510], [1, 30, 374, 532], [1, 30, 406, 508], [1, 34, 362, 540], [1, 42, 194, 620], [1, 44, 158, 630], [1, 52, 210, 614], [1, 52, 430, 486], [1, 58, 260, 594], [1, 58, 420, 494], [1, 76, 190, 618], [1, 78, 230, 604], [1, 78, 250, 596], [1, 78, 454, 460], [1, 84, 262, 590], [1, 84, 338, 550], [1, 90, 146, 628], [1, 90, 260, 590], [1, 90, 316, 562], [1, 92, 250, 594], [1, 92, 394, 510], [1, 98, 310, 564], [1, 100, 130, 630], [1, 100, 274, 582], [1, 100, 426, 482], [1, 102, 364, 530], [1, 108, 410, 494], [1, 114, 220, 602], [1, 122, 210, 604], [1, 130, 150, 620], [1, 130, 252, 586], [1, 130, 298, 564], [1, 130, 406, 492], [1, 130, 444, 458], [1, 134, 300, 562], [1, 134, 338, 540], [1, 150, 268, 574], [1, 150, 418, 476], [1, 154, 228, 590], [1, 154, 270, 572], [1, 172, 310, 546], [1, 186, 430, 452], [1, 190, 246, 572], [1, 190, 310, 540], [1, 194, 420, 458], [1, 202, 414, 460], [1, 210, 410, 460], [1, 220, 270, 550], [1, 220, 278, 546], [1, 222, 260, 554], [1, 230, 244, 558], [1, 230, 300, 530], [1, 242, 410, 444], [1, 250, 420, 430], [1, 252, 310, 514], [1, 254, 372, 470], [1, 260, 282, 526], [1, 260, 310, 510], [1, 260, 418, 426], [1, 266, 388, 450], [1, 270, 308, 506], [1, 286, 300, 502], [1, 286, 348, 470], [1, 298, 364, 450], [1, 310, 318, 476], [2, 4, 230, 609], [2, 6, 95, 644], [2, 6, 119, 640], [2, 6, 160, 631], [2, 6, 356, 545], [2, 7, 62, 648], [2, 7, 192, 622], [2, 7, 318, 568], [2, 7, 402, 512], [2, 9, 80, 646], [2, 10, 124, 639], [2, 12, 257, 598], [2, 12, 382, 527], [2, 14, 185, 624], [2, 15, 94, 644], [2, 15, 386, 524], [2, 16, 390, 521], [2, 18, 188, 623], [2, 18, 268, 593], [2, 20, 321, 566], [2, 20, 399, 514], [2, 23, 58, 648], [2, 24, 215, 614], [2, 26, 439, 480], [2, 28, 358, 543], [2, 30, 241, 604], [2, 31, 90, 644], [2, 31, 294, 580], [2, 32, 103, 642], [2, 32, 327, 562], [2, 32, 342, 553], [2, 32, 393, 518], [2, 36, 175, 626], [2, 36, 305, 574], [2, 36, 449, 470], [2, 38, 343, 552], [2, 39, 224, 610], [2, 39, 376, 530], [2, 41, 410, 504], [2, 47, 332, 558], [2, 48, 202, 617], [2, 49, 130, 636], [2, 49, 414, 500], [2, 50, 81, 644], [2, 50, 409, 504], [2, 52, 287, 582], [2, 54, 100, 641], [2, 54, 191, 620], [2, 56, 150, 631], [2, 56, 206, 615], [2, 57, 398, 512], [2, 58, 228, 607], [2, 64, 151, 630], [2, 65, 234, 604], [2, 65, 276, 586], [2, 68, 193, 618], [2, 68, 383, 522], [2, 70, 144, 631], [2, 70, 279, 584], [2, 70, 384, 521], [2, 70, 424, 489], [2, 71, 216, 610], [2, 71, 434, 480], [2, 72, 247, 598], [2, 72, 457, 458], [2, 73, 138, 632], [2, 73, 408, 502], [2, 74, 100, 639], [2, 78, 172, 623], [2, 78, 367, 532], [2, 80, 201, 614], [2, 80, 306, 569], [2, 80, 439, 474], [2, 82, 128, 633], [2, 82, 423, 488], [2, 84, 346, 545], [2, 84, 370, 529], [2, 86, 240, 599], [2, 87, 232, 602], [2, 87, 422, 488], [2, 89, 290, 576], [2, 94, 156, 625], [2, 94, 345, 544], [2, 94, 455, 456], [2, 95, 436, 474], [2, 97, 228, 602], [2, 97, 262, 588], [2, 97, 442, 468], [2, 100, 254, 591], [2, 100, 366, 529], [2, 102, 212, 607], [2, 102, 328, 553], [2, 103, 312, 562], [2, 104, 105, 634], [2, 104, 266, 585], [2, 104, 391, 510], [2, 105, 304, 566], [2, 106, 120, 631], [2, 108, 223, 602], [2, 111, 140, 626], [2, 111, 274, 580], [2, 111, 370, 524], [2, 111, 430, 476], [2, 112, 423, 482], [2, 114, 401, 500], [2, 116, 335, 546], [2, 122, 168, 617], [2, 124, 255, 586], [2, 124, 289, 570], [2, 124, 415, 486], [2, 127, 182, 612], [2, 128, 423, 478], [2, 129, 266, 580], [2, 130, 391, 504], [2, 134, 225, 596], [2, 134, 420, 479], [2, 135, 254, 584], [2, 135, 376, 514], [2, 136, 210, 601], [2, 136, 425, 474], [2, 137, 152, 618], [2, 137, 378, 512], [2, 138, 233, 592], [2, 138, 268, 577], [2, 138, 383, 508], [2, 142, 228, 593], [2, 142, 408, 487], [2, 145, 306, 556], [2, 146, 300, 559], [2, 146, 384, 505], [2, 150, 191, 604], [2, 151, 240, 586], [2, 152, 327, 542], [2, 152, 438, 457], [2, 156, 410, 481], [2, 158, 348, 527], [2, 160, 279, 566], [2, 161, 326, 540], [2, 162, 167, 608], [2, 162, 217, 592], [2, 162, 388, 497], [2, 162, 428, 463], [2, 164, 415, 474], [2, 167, 302, 552], [2, 168, 398, 487], [2, 169, 406, 480], [2, 170, 416, 471], [2, 172, 362, 513], [2, 174, 239, 580], [2, 175, 274, 564], [2, 178, 303, 548], [2, 180, 334, 529], [2, 183, 358, 512], [2, 184, 410, 471], [2, 185, 306, 544], [2, 185, 336, 526], [2, 188, 223, 582], [2, 190, 196, 591], [2, 190, 344, 519], [2, 192, 422, 457], [2, 193, 202, 588], [2, 194, 369, 500], [2, 196, 234, 575], [2, 196, 255, 566], [2, 196, 270, 559], [2, 200, 201, 586], [2, 201, 310, 536], [2, 201, 410, 464], [2, 202, 248, 567], [2, 202, 428, 447], [2, 206, 240, 569], [2, 208, 263, 558], [2, 210, 316, 529], [2, 212, 318, 527], [2, 214, 215, 576], [2, 214, 420, 449], [2, 215, 304, 534], [2, 215, 376, 486], [2, 216, 265, 554], [2, 217, 288, 542], [2, 225, 314, 524], [2, 228, 287, 538], [2, 232, 342, 503], [2, 239, 274, 540], [2, 239, 300, 526], [2, 240, 409, 446], [2, 241, 260, 546], [2, 241, 404, 450], [2, 252, 382, 463], [2, 254, 305, 516], [2, 254, 359, 480], [2, 257, 292, 522], [2, 258, 332, 497], [2, 258, 412, 433], [2, 262, 393, 448], [2, 265, 384, 454], [2, 266, 321, 500], [2, 268, 298, 513], [2, 268, 417, 422], [2, 270, 401, 436], [2, 271, 290, 516], [2, 272, 393, 442], [2, 273, 302, 508], [2, 274, 364, 465], [2, 280, 326, 489], [2, 281, 370, 456], [2, 282, 368, 457], [2, 284, 321, 490], [2, 288, 322, 487], [2, 289, 340, 474], [2, 292, 383, 438], [2, 296, 345, 466], [2, 302, 337, 468], [2, 302, 392, 423], [2, 304, 359, 450], [2, 304, 370, 441], [2, 306, 380, 431], [2, 311, 320, 474], [2, 312, 313, 478], [2, 316, 321, 470], [2, 329, 384, 410], [2, 340, 386, 399], [2, 344, 375, 406], [2, 348, 353, 422], [2, 351, 364, 410], [2, 361, 376, 390], ......, [300, 306, 342, 351], [300, 315, 324, 360], [301, 304, 320, 372], [302, 303, 308, 382], [302, 306, 335, 356], [302, 311, 326, 360], [304, 329, 330, 338], [305, 324, 334, 338], [305, 326, 330, 340], [306, 311, 320, 362], [308, 318, 337, 338], [308, 324, 325, 344], [309, 320, 328, 344], [310, 318, 329, 344], [310, 320, 326, 345], [313, 314, 330, 344]])

('651^3', 91, [[3, 6, 156, 648], [3, 18, 423, 585], [4, 76, 203, 644], [5, 213, 225, 634], [7, 35, 481, 548], [9, 195, 466, 551], [13, 369, 464, 501], [14, 35, 259, 637], [19, 90, 480, 548], [19, 114, 395, 597], [19, 324, 426, 548], [23, 252, 301, 615], [26, 145, 165, 645], [26, 228, 379, 594], [27, 162, 306, 624], [28, 35, 286, 632], [28, 140, 499, 530], [34, 74, 483, 546], [34, 179, 388, 596], [43, 81, 87, 650], [43, 134, 304, 626], [43, 180, 362, 606], [49, 119, 491, 538], [50, 333, 355, 579], [51, 111, 489, 540], [51, 147, 301, 626], [53, 122, 463, 559], [54, 171, 387, 597], [54, 270, 435, 558], [56, 70, 212, 643], [58, 299, 334, 596], [67, 342, 480, 500], [69, 219, 414, 579], [69, 290, 421, 561], [72, 207, 489, 531], [72, 241, 407, 579], [75, 142, 200, 642], [75, 186, 201, 639], [75, 298, 486, 512], [81, 156, 339, 615], [84, 402, 435, 504], [86, 187, 380, 598], [89, 195, 487, 534], [96, 131, 310, 624], [102, 107, 208, 642], [108, 354, 390, 555], [109, 387, 467, 486], [113, 180, 409, 585], [114, 426, 435, 486], [115, 306, 324, 596], [116, 339, 393, 559], [121, 278, 409, 569], [133, 343, 455, 518], [134, 151, 241, 635], [138, 309, 475, 515], [143, 202, 421, 575], [143, 289, 411, 564], [147, 252, 426, 564], [150, 261, 431, 559], [151, 241, 323, 608], [151, 254, 255, 621], [155, 278, 472, 526], [162, 346, 464, 507], [166, 374, 379, 548], [173, 304, 491, 499], [178, 407, 417, 507], [181, 297, 414, 557], [186, 279, 327, 597], [189, 273, 357, 588], [191, 195, 502, 513], [195, 216, 422, 568], [195, 302, 382, 570], [203, 252, 364, 588], [203, 384, 436, 504], [207, 291, 444, 537], [214, 300, 468, 515], [229, 233, 310, 605], [238, 329, 369, 561], [241, 293, 428, 541], [241, 297, 372, 569], [243, 408, 450, 468], [246, 360, 438, 507], [267, 278, 393, 559], [272, 293, 319, 583], [279, 403, 417, 488], [293, 301, 393, 546], [297, 333, 381, 540], [320, 401, 415, 475], [338, 411, 418, 456], [341, 372, 379, 507], [348, 418, 428, 435]])

('651^4', 1, [[240, 340, 430, 599]])

Treenewbee

cz1 发表于 2023-2-4 06:56
请问Treenewbee先生：
33199^3=2833^3+19081^3+30941^3=15187^3+24197^3+26647^3

(19081, 3, [7516, 9033, 17952])
(30941, 3, [7795, 23015, 25691])
(15187, 3, [6960, 7275, 14062])
(24197, 3, [2621, 16408, 21350])
(26647, 3, [14304, 19294, 20655])

Treenewbee

蔡家雄 发表于 2023-2-3 20:44
若 10 是质数 (a^2+b^2) 的原根，

且 质数(a^2+b^2) ≡ 17或33（mod  40），

10 也是 9^8+16^8=4338014017 的原根， 对
10 也是 1^4+24^4=331777 的原根， 错,405504
10 也是 19^4+24^4=462097 的原根，对
10 也是 7^16+8^16=314707907280257 的原根，对
10 也是 13^4+28^4=643217 的原根，对
10 也是 13^16+28^16=142735015363284129948737 的原根，错

蔡家雄

定义：形式 m*(m+1)/2 的数，叫：三角数。

3w+1 猜想

设 w 为三角数，且 3w+1 =完全平方数，

例 w= 1, 21, 120, ...... , 使命题成立，求 w 的通解公式。

yangchuanju

yangchuanju 发表于 2023-2-4 06:54
蔡家雄两方程有解，最小解是m=353，非651也！
m^3=a^3+b^3+c^3+d^3
m^4=e^4+f^4+g^4+ h^4

重要更正：
491楼给出的方程解有误，它们不是方程组
m^3=a^3+b^3+c^3+d^3
m^4=e^4+f^4+g^4+ h^4
的解，而是
m=a^3+b^3+c^3+d^3
m^4=e^4+f^4+g^4+ h^4
的解，特此更正！

蔡家雄

求解方程：

m^3=a^3+b^3+c^3+d^3

m^5=e^5+f^5+g^5+ h^5

85359^3= a^3+b^3+c^3+d^3

85359^5= 85282^5+28969^5+3183^5+55^5


求解方程：

m^4=a^4+b^4+c^4+d^4

m^5=e^5+f^5+g^5+ h^5

85359^4= a^4+b^4+c^4+d^4

85359^5= 85282^5+28969^5+3183^5+55^5

蔡家雄

若 10 是质数 (a^2+b^2) 的原根，

且 质数(a^2+b^2) ≡ 17或33（mod  40），

则 10 也是质数 (a^(2^n)+b^(2^n)) 的原根。


已知 10 是 9^2+16^2=337 的原根，

判断 10 也是 9^8+16^8=4338014017 的原根，


已知 10 是 1^2+24^2=577 的原根，

判断 10 也是 1^4+24^4=331777 的原根，


已知 10 是 19^2+24^2=937 的原根，

判断 10 也是 19^4+24^4=462097 的原根，


已知 10 是 7^2+8^2=113 的原根，

判断 10 也是 7^16+8^16=314707907280257 的原根，


已知 10 是 13^2+28^2=953 的原根，

判断 10 也是 13^4+28^4=643217 的原根，

判断 10 也是 13^16+28^16=142735015363284129948737 的原根，

蔡家雄

Treenewbee 的有问题

10 也是 9^8+16^8=4338014017 的原根， 对
10 也是 1^4+24^4=331777 的原根， 错,405504  ( 这个判断有问题 ，为何会是 405504 )
10 也是 19^4+24^4=462097 的原根，对
10 也是 7^16+8^16=314707907280257 的原根，对
10 也是 13^4+28^4=643217 的原根，对
10 也是 13^16+28^16=142735015363284129948737 的原根，错