n^2=a^2+b^2-1

王守恩 · 发表于 2025-12-20 10:54:56

$n=a^2+b^2-c^3$ ——n 能跑遍所有自然数吗？——可有通项公式？

{{n -> 0, a -> 2, b -> 2, c -> 2}}, {{n -> 1, a -> 7,  b -> 4,  c -> 4}}, {{n -> 2, a -> 5, b -> 2, c -> 3}}, {{n -> 3, a -> 8, b -> 8, c -> 5}}, {{n -> 4, a -> 8, b -> 2, c -> 4}},
{{n -> 5, a -> 3, b -> 2, c -> 2}}, {{n -> 6, a -> 18, b -> 5, c -> 7}}, {{n -> 7, a -> 2, b -> 2, c -> 1}}, {{n -> 8, a -> 6, b -> 6, c -> 4}}, {{n -> 9, a -> 8, b -> 3, c -> 4}},
{{n -> 10, a -> 3, b -> 3, c -> 2}},{{n -> 11,a -> 10, b -> 6, c -> 5}}, {{n -> 12, a -> 3, b -> 2, c -> 1}},{{n ->13, a -> 6, b -> 2, c -> 3}}, {{n ->14, a -> 5, b -> 4, c -> 3}},
{{n -> 15, a ->35,b ->11,c ->11}}, {{n -> 16, a -> 8, b -> 4, c -> 4}}, {{n -> 17, a -> 3, b -> 3, c -> 1}},{{n ->18, a -> 6, b -> 3, c -> 3}}, {{n ->19, a -> 4, b -> 2, c -> 1}},
{{n -> 20, a -> 9, b -> 8, c -> 5}}, {{n -> 21, a -> 5, b -> 2, c -> 2}}, {{n -> 22, a ->14, b ->13, c ->7}}, {{n ->23, a -> 5, b -> 5, c -> 3}}, {{n -> 24, a -> 4, b -> 3, c -> 1}},
{{n -> 25, a -> 6, b -> 4, c -> 3}}, {{n -> 26, a -> 7, b -> 2, c -> 3}}, {{n -> 27, a ->17, b -> 9, c -> 7}}, {{n -> 28, a -> 5, b -> 2, c -> 1}}, {{n ->29, a ->14, b -> 7, c -> 6}},
{{n -> 30, a ->18, b -> 7, c -> 7}}, {{n -> 31, a -> 7, b -> 3, c -> 3}}, {{n -> 32, a -> 6, b -> 2, c -> 2}}, {{n -> 33, a -> 5, b -> 3, c -> 1}}, {{n -> 34, a -> 6, b -> 5, c -> 3}},
{{n -> 35, a ->12, b -> 4, c -> 5}}, {{n -> 36, a -> 8, b -> 6, c -> 4}}, {{n -> 37, a -> 6, b -> 3, c -> 2}}, {{n -> 38, a -> 7, b -> 4, c -> 3}}, {{n -> 39, a -> 6, b -> 2, c -> 1}},
{{n -> 40, a -> 5, b -> 4, c -> 1}}, {{n -> 41, a -> 8, b -> 2, c -> 3}}, {{n -> 42, a -> 5, b -> 5, c -> 2}}, {{n -> 43, a -> 19, b ->5, c -> 7}}, {{n -> 44, a -> 6, b -> 3, c -> 1}},
{{n -> 45, a -> 7, b -> 2, c -> 2}}, {{n -> 46, a -> 8, b -> 3, c -> 3}}, {{n -> 47, a -> 7, b -> 5, c -> 3}}, {{n -> 48, a -> 13, b -> 2, c ->5}}, {{n -> 49, a -> 5, b -> 5, c -> 1}},
{{n -> 50, a -> 7, b -> 3, c -> 2}}, {{n -> 51, a -> 6, b -> 4, c -> 1}}, {{n -> 52, a -> 7, b -> 2, c -> 1}}, {{n -> 53, a -> 8, b -> 4, c -> 3}}, {{n -> 54, a ->19, b -> 6, c -> 7}},
{{n -> 55, a ->12, b -> 6, c -> 5}}, {{n ->56, a ->10, b -> 9, c -> 5}}, {{n -> 57, a -> 7, b -> 3, c -> 1}}, {{n -> 58, a -> 9, b -> 2, c -> 3}}, {{n -> 59, a ->28, b -> 2, c -> 9}},
{{n -> 60, a -> 8, b -> 2, c -> 2}}, {{n -> 61, a ->11, b -> 2, c -> 4}}, {{n -> 62, a -> 8, b -> 5, c -> 3}}, {{n -> 63, a -> 9, b -> 3, c -> 3}}, {{n -> 64, a -> 6, b -> 6, c -> 2}},
{{n -> 65, a -> 8, b -> 3, c -> 2}}, {{n -> 66, a -> 7, b -> 5, c -> 2}}, {{n -> 67, a -> 8, b -> 2, c -> 1}}, {{n -> 68, a ->12, b -> 7, c -> 5}},{{n -> 69, a ->13, b -> 5, c -> 5}},
{{n -> 70, a -> 9, b -> 4, c -> 3}}, {{n -> 71, a -> 7, b -> 7, c -> 3}}, {{n -> 72, a -> 8, b -> 3, c -> 1}}, {{n -> 73, a -> 8, b -> 6, c -> 3}}, {{n -> 74, a -> 13, b ->11, c -> 6}},
{{n -> 75, a -> 14, b -> 2, c -> 5}},{{n ->76, a -> 16, b -> 6, c -> 6}},{{n -> 77, a -> 9, b -> 2, c -> 2}}, {{n -> 78, a ->15, b ->14, c ->7}}, {{n -> 79,  a -> 9, b -> 5, c -> 3}},
{{n -> 80, a -> 14, b -> 3, c -> 5}}, {{n -> 81, a -> 8, b -> 5, c -> 2}}, {{n -> 82, a -> 9, b -> 3, c -> 2}}, {{n -> 83, a -> 12, b -> 8, c -> 5}}, {{n -> 84, a -> 9, b -> 2, c -> 1}},
{{n -> 85, a -> 10, b -> 7, c -> 4}}, {{n -> 86, a -> 8, b -> 7, c -> 3}}, {{n -> 87, a -> 14, b -> 4, c -> 5}}, {{n -> 88, a -> 8, b -> 5, c -> 1}}, {{n -> 89, a -> 9, b -> 3, c -> 1}},
{{n -> 90, a ->  9,  b -> 6, c -> 3}}, {{n -> 91, a ->28, b -> 6, c -> 9}}, {{n -> 92, a -> 8, b -> 6, c -> 2}}, {{n -> 93, a -> 13, b -> 7, c -> 5}},{{n ->94, a ->45, b ->38, c ->15}},
{{n -> 95, a -> 11, b -> 1, c -> 3}}, {{n -> 96, a ->10, b -> 2, c -> 2}}, {{n -> 97, a -> 7, b -> 7, c -> 1}}, {{n -> 98, a -> 11, b -> 2, c -> 3}}, {{n -> 99, a -> 8, b -> 6, c -> 1}},

更—— $n=a^2+b^2-c^4$ ——n 能跑遍所有自然数吗？

王守恩 · 发表于 2025-12-20 17:48:09

$n^2=a^2+b^2-0$——A009003——Hypotenuse numbers (squares are sums of 2 nonzero squares).
5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115,

$n^2=a^2+b^2-1$——A050796 Numbers n such that n^2 + 1 is expressible as the sum of two nonzero squares in at least one way (the trivial solution n^2 + 1 = n^2 + 1^2 is not counted).
1, 7, 8, 12, 13, 17, 18, 21, 22, 23, 27, 28, 30, 31, 32, 33, 34, 37, 38, 41, 42, 43, 44, 46, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 64, 67, 68, 70, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 86, 87, 88, 89, 91, 92, 93, 96,

$n^2=a^2+b^2-2$——A274595——Numbers n such that n^2 + 2 is the sum of two nonzero squares.
0, 4, 12, 24, 40, 48, 60, 68, 72, 84, 104, 112, 132, 140, 144, 148, 176, 180, 192, 204, 216, 220, 252, 264, 276, 284, 312, 320, 324, 364, 372, 384, 392, 396, 408, 420, 428, 444, 456, 468, 472, 480, 528, 544, 588, 600,

$n^2=a^2+b^2-3$——OEIS还没有。
{7, 17, 25, 31, 41, 49, 55, 71, 95, 97, 119, 127, 137, 143, 151, 161, 169, 199, 217, 223, 233, 239, 241, 257, 263, 265, 287, 295, 305, 311, 329, 335, 337, 353, 377, 385, 391, 407, 409, 431, 449, 455, 463, 473, 487, 497}

$n^2=a^2+b^2-4$——OEIS还没有。
{9, 11, 14, 16, 19, 21, 23, 24, 25, 26, 29, 31, 34, 36, 39, 41, 42, 43, 44, 46, 49, 51, 53, 54, 55, 56, 59, 60, 61, 62, 63, 64, 66, 68, 69, 71, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 96, 99, 100, 101, 104}

$n^2=a^2+b^2-5$——OEIS还没有。
{0, 6, 12, 16, 20, 30, 36, 42, 48, 56, 66, 70, 72, 78, 90, 92, 96, 106, 110, 114, 124, 126, 132, 156, 160, 162, 164, 168, 180, 182, 198, 204, 210, 216, 218, 228, 232, 240, 246, 252, 258, 272, 286, 288, 294, 306, 322, 324, 326}

$n^2=a^2+b^2-6$——OEIS还没有。
{2, 10, 14, 22, 38, 46, 58, 74, 82, 86, 94, 98, 110, 122, 142, 154, 166, 170, 178, 182, 206, 214, 218, 226, 242, 250, 254, 262, 266, 310, 322, 334, 338, 350, 362, 374, 382, 394, 406, 410, 418, 434, 478, 494, 506, 518, 530, 542}

$n^2=a^2+b^2-7$——OEIS还没有。
{1, 5, 11, 15, 17, 29, 31, 33, 43, 47, 59, 61, 65, 67, 69, 81, 85, 93, 95, 99, 113, 125, 127, 129, 131, 143, 159, 171, 177, 181, 187, 191, 193, 195, 197, 213, 223, 225, 227, 239, 241, 253, 261, 267, 271, 285, 289, 305, 309, 319, 323}

$n^2=a^2+b^2-8$——OEIS还没有。
{0, 3, 8, 9, 15, 19, 21, 24, 33, 35, 37, 48, 51, 53, 55, 57, 63, 73, 80, 81, 87, 89, 96, 99, 109, 111, 117, 120, 123, 125, 129, 135, 136, 141, 143, 144, 147, 153, 163, 165, 168, 177, 189, 195, 201, 207, 208, 213, 217, 219, 224, 235, 243}

$n^2=a^2+b^2-9$——OEIS还没有。
{11, 14, 16, 19, 21, 22, 24, 26, 28, 29, 31, 34, 36, 37, 39, 41, 44, 46, 49, 50, 51, 54, 55, 56, 59, 61, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76, 79, 80, 81, 84, 86, 89, 90, 91, 92, 93, 94, 96, 97, 99, 101, 102, 104, 106, 107, 109, 111, 114}

王守恩 · 发表于 2025-12-20 18:31:04

$n^2=a^2+b^2+0$——A009003——Hypotenuse numbers (squares are sums of 2 nonzero squares).
5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122,

$n^2=a^2+b^2+1$——A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.
3, 9, 17, 19, 33, 35, 51, 73, 81, 99, 105, 129, 145, 147, 161, 163, 179, 195, 201, 233, 243, 273, 289, 291, 297, 339, 361, 387, 393, 451, 465, 467, 483, 489, 513, 521, 577, 579, 585, 611, 627, 649, 675, 721, 723, 739, 777, 801, 809, 819,

$n^2=a^2+b^2+2$——OEIS还没有。
6, 10, 14, 22, 26, 30, 34, 42, 50, 58, 62, 78, 82, 86, 90,  98, 106, 114, 118, 126, 138, 142, 146,  162, 170, 174, 182, 186, 190, 198, 206, 210, 218, 222, 226, 230, 250, 254, 266, 274, 278, 282, 286, 306, 310, 314, 330, 334, 338, 342, 358,

$n^2=a^2+b^2+3$——OEIS还没有。
4, 8, 10, 14, 20, 22, 26, 32, 34,40, 44, 46, 52, 56, 58, 64, 68, 74, 80, 86, 88, 92, 94, 98, 100, 110, 112, 118, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 158, 164, 172, 178, 184, 190, 194, 202, 206, 208, 212, 218, 220, 230, 238, 242,

$n^2=a^2+b^2+4$——OEIS还没有。
3, 6, 7, 11, 15, 18, 27, 34, 38, 39, 43, 47, 51, 63, 66, 70, 83, 87, 99, 102, 111, 115, 119, 123, 146, 147, 151, 155, 162, 171, 183, 195, 198, 210, 223, 227, 231, 243, 258, 259, 263, 267, 279, 290, 291, 294, 315, 322, 326, 335, 351, 358, 363,

$n^2=a^2+b^2+5$——OEIS还没有。
3, 5, 11, 13, 19, 21, 27, 35, 43, 45, 53, 61, 69, 75, 77, 83, 85, 91, 93, 101, 107, 109, 115, 131, 133, 141, 155, 157, 163, 165, 171, 173, 179, 187, 189, 195, 197, 203, 221, 229, 243, 245, 251, 253, 259, 261, 269, 277, 283, 291, 299, 307, 309,

$n^2=a^2+b^2+6$——OEIS还没有。
4, 8, 16, 20, 28, 32, 40, 44, 56, 64, 68, 76, 88, 92, 112, 116, 124, 136, 140, 148, 152, 164, 184, 188, 200, 212, 220, 224, 236, 244, 248, 256, 260,268, 284, 296, 304, 316, 320, 328, 332, 340, 344, 352, 364, 368, 388, 392, 400, 412, 416, 424,

$n^2=a^2+b^2+7$——OEIS还没有。
4, 5, 6, 9, 12, 13, 15, 18, 22, 23, 24, 27, 31, 32, 33, 36, 39, 45, 48, 50, 51, 54, 57, 58, 59, 60, 66, 69, 75, 76, 81, 85, 86, 90, 93, 94, 96, 102, 104, 108, 113, 117, 120, 123, 129, 130, 131, 132, 135, 138, 139, 149, 153, 156, 157, 159, 162, 166,

$n^2=a^2+b^2+8$——OEIS还没有。
4, 5, 7, 9, 11, 12, 17, 19, 20, 21, 23, 25, 28, 29, 31, 35, 37, 39, 44, 45, 49, 51,  52, 53, 60, 63,  65, 67, 68, 69, 73, 75, 81, 84, 87,  89, 91, 93, 100, 101, 103, 107, 115, 116, 117, 119, 121, 123, 124, 129, 131, 133, 135, 137, 143, 145, 147, 149,

$n^2=a^2+b^2+9$——OEIS还没有。
5, 7, 9, 13, 23, 27, 29, 37, 51, 55, 57, 61, 71, 77, 99, 101, 103, 105, 119, 125, 133, 153, 167, 197, 199, 205, 215, 219, 229, 243, 247, 253, 293, 295, 297, 315, 317, 343, 359, 387, 389, 391, 397, 407, 413, 435, 439, 441, 455, 461, 483, 485, 487,

王守恩 · 发表于 2025-12-21 13:09:50

$n^k=a^2 + b^2 - c^2$——这个可以有！

$n^3=a^2 + b^2 - c^2$
$00^3=0^2+1^2-1^2$
$01^3=1^2+2^2-2^2$
$02^3=4^2+1^2-3^2$
$03^3=12^2+2^2-11^2$
$04^3=32^2+1^2-31^2$
$05^3=61^2+2^2-60^2$
$06^3=108^2+1^2-107^2$
$07^3=170^2+2^2-169^2$
$08^3=256^2+1^2-255^2$
$09^3=363^2+2^2-362^2$
$10^3=500^2+1^2-499^2$
$11^3=664^2+2^2-663^2$
$12^3=864^2+1^2-863^2$
$13^3=1097^2+2^2-1096^2$
$14^3=1372^2+1^2-1371^2$
$15^3=1686^2+2^2-1685^2$

$n^4=a^2 + b^2 - c^2$
$00^4=0^2+1^2-1^2$
$01^4=1^2+2^2-2^2$
$02^4=8^2+1^2-7^2$
$03^4=39^2+2^2-38^2$
$04^4=128^2+1^2-127^2$
$05^4=311^2+2^2-310^2$
$06^4=648^2+1^2-647^2$
$07^4=1199^2+2^2-1198^2$
$08^4=2048^2+1^2-2047^2$
$09^4=3279^2+2^2-3278^2$
$10^4=5000^2+1^2-4999^2$
$11^4=7319^2+2^2-7318^2$
$12^4=10368^2+1^2-10367^2$
$13^4=14279^2+2^2-14278^2$
$14^4=19208^2+1^2-19207^2$
$15^4=25311^2+2^2-25310^2$

$n^5=a^2 + b^2 - c^2$
$00^5=0^2+1^2-1^2$
$01^5=1^2+2^2-2^2$
$02^5=16^2+1^2-15^2$
$03^5=120^2+2^2-119^2$
$04^5=512^2+1^2-511^2$
$05^5=1561^2+2^2-1560^2$
$06^5=3888^2+1^2-3887^2$
$07^5=8402^2+2^2-8401^2$
$08^5=16384^2+1^2-16383^2$
$09^5=29523^2+2^2-29522^2$
$10^5=50000^2+1^2-49999^2$
$11^5=80524^2+2^2-80523^2$
$12^5=124416^2+1^2-124415^2$
$13^5=185645^2+2^2-185644^2$
$14^5=268912^2+1^2-268911^2$
$15^5=379686^2+2^2-379685^2$

$n^6=a^2 + b^2 - c^2$
$00^6=0^2+1^2-1^2$
$01^6=1^2+2^2-2^2$
$02^6=32^2+1^2-31^2$
$03^6=363^2+2^2-362^2$
$04^6=2048^2+1^2-2047^2$
$05^6=7811^2+2^2-7810^2$
$06^6=23328^2+1^2-23327^2$
$07^6=58823^2+2^2-58822^2$
$08^6=131072^2+1^2-131071^2$
$09^6=265719^2+2^2-265718^2$
$10^6=500000^2+1^2-499999^2$
$11^6=885779^2+2^2-885778^2$
$12^6=1492992^2+1^2-1492991^2$
$13^6=2413403^2+2^2-2413402^2$
$14^6=3764768^2+1^2-3764767^2$
$15^6=5695311^2+2^2-5695310^2$

王守恩 · 发表于 2025-12-21 13:13:30

$n^k=a^2 + b^2 - c^2$——这个有点"高级"！

$n^3=a^2 + b^2 - c^2$
$00^3=0^2+1^2-1^2$
$01^3=1^2+0^2-0^2$
$02^3=4^2+1^2-3^2$
$03^3=12^2+2^2-11^2$
$04^3=28^2+3^2-27^2$
$05^3=55^2+4^2-54^2$
$06^3=96^2+5^2-95^2$
$07^3=154^2+6^2-153^2$
$08^3=232^2+7^2-231^2$
$09^3=333^2+8^2-332^2$
$10^3=460^2+9^2-459^2$
$11^3=616^2+10^2-615^2$
$12^3=804^2+11^2-803^2$
$13^3=1027^2+12^2-1026^2$
$14^3=1288^2+13^2-1287^2$
$15^3=1590^2+14^2-1589^2$

$n^4=a^2 + b^2 - c^2$
$00^4=0^2+1^2-1^2$
$01^4=1^2+0^2-0^2$
$02^4=8^2+1^2-7^2$
$03^4=39^2+2^2-38^2$
$04^4=124^2+3^2-123^2$
$05^4=305^2+4^2-304^2$
$06^4=636^2+5^2-635^2$
$07^4=1183^2+6^2-1182^2$
$08^4=2024^2+7^2-2023^2$
$09^4=3249^2+8^2-3248^2$
$10^4=4960^2+9^2-4959^2$
$11^4=7271^2+10^2-7270^2$
$12^4=10308^2+11^2-10307^2$
$13^4=14209^2+12^2-14208^2$
$14^4=19124^2+13^2-19123^2$
$15^4=25215^2+14^2-25214^2$

$n^5=a^2 + b^2 - c^2$
$00^5=0^2+1^2-1^2$
$01^5=1^2+0^2-0^2$
$02^5=16^2+1^2-15^2$
$03^5=120^2+2^2-119^2$
$04^5=508^2+3^2-507^2$
$05^5=1555^2+4^2-1554^2$
$06^5=3876^2+5^2-3875^2$
$07^5=8386^2+6^2-8385^2$
$08^5=16360^2+7^2-16359^2$
$09^5=29493^2+8^2-29492^2$
$10^5=49960^2+9^2-49959^2$
$11^5=80476^2+10^2-80475^2$
$12^5=124356^2+11^2-124355^2$
$13^5=185575^2+12^2-185574^2$
$14^5=268828^2+13^2-268827^2$
$15^5=379590^2+14^2-379589^2$

$n^6=a^2 + b^2 - c^2$
$00^6=0^2+1^2-1^2$
$01^6=1^2+0^2-0^2$
$02^6=32^2+1^2-31^2$
$03^6=363^2+2^2-362^2$
$04^6=2044^2+3^2-2043^2$
$05^6=7805^2+2^4-7804^2$
$06^6=23316^2+5^2-23315^2$
$07^6=58807^2+6^2-58806^2$
$08^6=131048^2+7^2-131047^2$
$09^6=265689^2+8^2-265688^2$
$10^6=499960^2+9^2-499959^2$
$11^6=885731^2+10^2-885730^2$
$12^6=1492932^2+11^2-1492931^2$
$13^6=2413333^2+12^2-2413332^2$
$14^6=3764684^2+13^2-3764683^2$
$15^6=5695215^2+14^2-5695214^2$

王守恩 · 发表于 2025-12-21 14:41:47

northwolves 发表于 2025-12-17 23:09
更狠一点。n, a, b, c 是不同的数。$n^2=a^2+b^2-c^k$ k(固定)
c=1 or r^2

这个题只有你能做——我做不了。谢谢！

$n=a^2+b^2-c^2$, a,b,c是正整数。 a(n)=a+b+c的最小数。

a(1)=1+1+1=3,
a(2)=3+3+4=10,
a(3)=4+6+7=17,
a(4)=1+2+1=4,
a(5)=4+5+6=15,
a(6)=1+3+2=6,
a(7)=2+2+1=5,
a(8)=1+4+3=8,
a(9)=1+3+1=5,
a(10)=1+5+4=10,
a(11)=2+4+3=9,
a(12)=2+3+1=6,
a(13)=1+4+2=7,
a(14)=3+3+2=8,

creasson · 发表于 2025-12-21 15:53:45

$n^k=a^2+b^2-c^2$始终有如下有理解：
\[ a = \frac{u^2 n^k-v^2 n^k+1}{2 u}, \quad b =  v n^k, \quad c = \frac{u^2 n^k+v^2 n^k-1}{2 u} \]
例如，当 $n=2m$时，可取$ u = v = \frac{1}{2} $ ，即有
\[1^2 + \left(\frac{n^k}{2}\right)^2-\left(\frac{n^k}{2} - 1 \right)^2 =n^k \]
当 $ n = 2m + 1 $ 时, 可取$ u =4 n^k-1,  v = 2 $ ，即有
\[\left(\frac{4 n^{2 k}-n^k-1}{2} \right)^2 +  \left(2 n^k\right)^2 -\left(\frac{4 n^{2 k}-n^k+1}{2} \right)^2=n^k \]

王守恩 · 发表于 2025-12-22 14:41:34

$n=a^2+b^2-c^4$ ——n 能跑遍所有自然数吗？——谢谢各位！

譬如：n={6, 11, 14, 22, 27, 30, 38, 43, 46, 54, 59, 62, 70, 75, 78, 86, 91, 94, 102, 107, 110, 118, 123, 126, 134, 139, 142, 150, 155, 158, 166, 171, 174, 182, 187, 190, 198, 203, 206, 214}会有解吗？

0=20^2+15^2-5^4
1=1^2+1^2-1^4
2=3^2+3^2-2^4
3=22^2+12^2-5^4
4=2^2+1^2-1^4
5=15^2+6^2-4^4
6=
7=2^2+2^2-1^4
8=8^2+5^2-3^4
9=3^2+1^2-1^4
10=5^2+1^2-2^4
11=
12=3^2+2^2-1^4
13=5^2+2^2-2^4
14=
15=24^2+8^2-5^4
16=4^2+1^2-1^4
17=3^2+3^2-1^4
18=5^2+3^2-2^4
19=4^2+2^2-1^4
20=10^2+1^2-3^4
21=6^2+1^2-2^4
22=
23=10^2+2^2-3^4
24=6^2+2^2-2^4
25=5^2+1^2-1^4
26=99^2+15^2-10^4
27=
28=5^2+2^2-1^4
29=6^2+3^2-2^4
30=

northwolves · 发表于 2025-12-22 17:09:09

王守恩发表于 2025-12-22 14:41
$n=a^2+b^2-c^4$ ——n 能跑遍所有自然数吗？——谢谢各位！

譬如：n={6, 11, 14, 22, 27, 30, 38, 4 ...

m = Union@Table[Mod[k[[1]]^2 + k[[2]]^2 - k[[3]]^4, 16], {k, Tuples[Range[0, 15], {3}]}]

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{0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 15}

{6, 11, 14, 22, 27, 30, 38, 43, 46, 54, 59, 62, 70, 75, 78, 86, 91, 94, 102, 107, 110, 118, 123, 126, 134, 139, 142, 150, 155, 158, 166, 171, 174, 182, 187, 190, 198, 203, 206, 214} Mod 16={6, 11, 14}

王守恩 · 发表于 2025-12-22 18:10:52

northwolves 发表于 2025-12-22 17:09
{0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 15}

譬如：n={6, 11, 14, 22, 27, 30, 38, 43, 46, 54, 59, 62, 70, 75, 78, 86, 91, 94, 102, 107, 110, 118, 123, 126, 134, 139, 142, 150, 155, 158, 166, 171, 174, 182, 187, 190, 198, 203, 206, 214}会有解吗？

Table[16*Floor[(n - 1)/3] + {6, 11, 14}[[Mod[n, 3, 1]]], {n, 50}]

Table[(48 n - 15 Cos[(2 n Pi)/3] + Sqrt[3] Sin[(2 n Pi)/3] - 3)/9, {n, 50}]

关键是：为什么说这些数是无解的？2, 这些数还漏了几个——63, 143, .... 3, 因为OEIS没有这些数，困难就多了。我这个电脑只能出来0——200。后面的验算不了。

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[投票] n^2=a^2+b^2-1

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