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

王守恩

northwolves 发表于 2025-12-27 13:12
易知下面两个方程对于任意正整数n都有解：

$ n=a^1+b^2-c^3$

$ n=a^1+b^2-c^3$——这个还是容易的。
$ 1=0^1+3^2-2^3$
$ 2=1^1+3^2-2^3$
$ 3=2^1+3^2-2^3$
$ 4=3^1+3^2-2^3$
$ 5=4^1+3^2-2^3$
$ 6=5^1+3^2-2^3$
$ 7=6^1+3^2-2^3$
$ 8=7^1+3^2-2^3$
$ 9=8^1+3^2-2^3$

$ n=a^2+b^3-c^4$——这个有"容易"的吗？

对每个n来说：有1个解——就有无数个解。譬如——n=1。——就有无数个解。

Solve[{1 == a^2 + b^3 - c^4, 0 < a, 0 < b, 0 < c < 50}, {c, b, a}, Integers]

{{c -> 1, b -> 1, a -> 1}, {c -> 2, b -> 1, a -> 4}, {c -> 2, b -> 2, a -> 3}, {c -> 3, b -> 1, a -> 9}, {c -> 4, b -> 1, a -> 16}, {c -> 5, b -> 1, a -> 25},
{c -> 6, b -> 1, a -> 36}, {c -> 7, b -> 1, a -> 49}, {c -> 8, b -> 1, a -> 64}, {c -> 8, b -> 16, a -> 1}, {c -> 9, b -> 1, a -> 81}, {c -> 10, b -> 1, a -> 100},
{c -> 11, b -> 1, a -> 121}, {c -> 12, b -> 1, a -> 144}, {c -> 12, b -> 16, a -> 129}, {c -> 13, b -> 1, a -> 169}, {c -> 14, b -> 1, a -> 196}, {c -> 15, b -> 1, a -> 225},
{c -> 16, b -> 1, a -> 256}, {c -> 16, b -> 8, a -> 255}, {c -> 16, b -> 26, a -> 219}, {c -> 16, b -> 37, a -> 122}, {c -> 17, b -> 1, a -> 289}, {c -> 18, b -> 1, a -> 324},
{c -> 19, b -> 1, a -> 361}, {c -> 20, b -> 1, a -> 400}, {c -> 21, b -> 1, a -> 441}, {c -> 22, b -> 1, a -> 484}, {c -> 23, b -> 1, a -> 529}, {c -> 24, b -> 1, a -> 576},
{c -> 25, b -> 1, a -> 625}, {c -> 26, b -> 1, a -> 676}, {c -> 27, b -> 1, a -> 729}, {c -> 27, b -> 81, a -> 1}, {c -> 28, b -> 1, a -> 784}, {c -> 29, b -> 1, a -> 841},
{c -> 30, b -> 1, a -> 900}, {c -> 31, b -> 1, a -> 961}, {c -> 32, b -> 1, a -> 1024}, {c -> 33, b -> 1, a -> 1089}, {c -> 33, b -> 81, a -> 809}, {c -> 34, b -> 1, a -> 1156},
{c -> 35, b -> 1, a -> 1225}, {c -> 36, b -> 1, a -> 1296}, {c -> 36, b -> 106, a -> 699}, {c -> 37, b -> 1, a -> 1369}, {c -> 38, b -> 1,  a -> 1444}, {c -> 38, b -> 92, a -> 1143},
{c -> 39, b -> 1, a -> 1521}, {c -> 40, b -> 1, a -> 1600}, {c -> 40, b -> 100, a -> 1249}, {c -> 41, b -> 1, a -> 1681}, {c -> 42, b -> 1,  a -> 1764}, {c -> 43, b -> 1, a -> 1849},
{c -> 44, b -> 1, a -> 1936}, {c -> 45, b -> 1, a -> 2025}, {c -> 46, b -> 1, a -> 2116}, {c -> 47, b -> 1, a -> 2209}, {c -> 48, b -> 1, a -> 2304}, {c -> 49, b -> 1, a -> 2401}}

只能每个n就来1个解——前200个n——注意——x = 4; y = 3; z = 2;——是可以改的。

x = 4; y = 3; z = 2; Table[Catch@Do[For[b = Floor[(n + c^x)^(1/y)], b > 0, b--, a = (n + c^x - b^y)^(1/z); If[IntegerQ@a && a > 0, Throw@{n, c, b, a}]], {c, \[Infinity]}], {n, 200}]

{{1, 1, 1, 1}, {2, 69, 283, 44}, {3, 7, 12, 26}, {4, 1, 1, 2}, {5, 46, 113, 1742}, {6, 13, 7, 168}, {7, 5, 7, 17}, {8, 1, 2, 1}, {9, 1, 1, 3}, {10, 2, 1, 5},
{11, 1, 2, 2}, {12, 2, 3, 1}, {13, 4, 5, 12}, {14, 27, 79, 196}, {15, 2, 3, 2}, {16, 1, 2, 3}, {17, 2, 2, 5}, {18, 28, 49, 705}, {19, 3, 4, 6}, {20, 2, 3, 3},
{21, 2, 1, 6}, {22, 37, 103, 884}, {23, 1, 2, 4}, {24, 4, 6, 8}, {25, 1, 1, 5}, {26, 11, 23, 50}, {27, 1, 3, 1}, {28, 2, 2, 6}, {29, 5, 5, 23}, {30, 1, 3, 2},
{31, 5, 8, 12}, {32, 1, 2, 5}, {33, 4, 4, 15}, {34, 2, 1, 7}, {35, 1, 3, 3}, {36, 1, 1, 6}, {37, 8, 13, 44}, {38, 4, 5, 13}, {39, 6, 11, 2}, {40, 8, 10, 56},
{41, 2, 2, 7}, {42, 1, 3, 4}, {43, 1, 2, 6}, {44, 6, 11, 3}, {45, 3, 5, 1}, {46, 3, 3, 10}, {47, 2, 3, 6}, {48, 3, 5, 2}, {49, 1, 1, 7}, {50, 18, 25, 299},
{51, 1, 3, 5}, {52, 2, 4, 2}, {53, 3, 5, 3}, {54, 6, 5, 35}, {55, 9, 18, 28}, {56, 1, 2, 7}, {57, 2, 4, 3}, {58, 6, 9, 25}, {59, 5, 2, 26}, {60, 2, 3, 7},
{61, 318, 1017, 95782}, {62, 1, 3, 6}, {63, 8, 15, 28}, {64, 1, 4, 1}, {65, 4, 5, 14}, {66, 2, 1, 9}, {67, 1, 4, 2}, {68, 13, 17, 154}, {69, 3, 5, 5}, {70, 117, 555, 4054},
{71, 1, 2, 8}, {72, 1, 4, 3}, {73, 2, 4, 5}, {74, 6, 1, 37}, {75, 1, 3, 7}, {76, 4, 2, 18}, {77, 15, 37, 7}, {78, 5, 3, 26}, {79, 1, 4, 4}, {80, 3, 5, 6},
{81, 1, 1, 9}, {82, 17, 43, 64}, {83, 3, 4, 10}, {84, 2, 4, 6}, {85, 2, 1, 10}, {86, 7, 11, 34}, {87, 49, 54, 2368}, {88, 1, 4, 5}, {89, 3, 1, 13}, {90, 1, 3, 8},
{91, 4, 7, 2}, {92, 2, 3, 9}, {93, 3, 5, 7}, {94, 4, 5, 15}, {95, 4, 3, 18}, {96, 3, 2, 13}, {97, 2, 4, 7}, {98, 22, 17, 479}, {99, 1, 4, 6}, {100, 1, 1, 10},
{101, 205, 1205, 4051}, {102, 10, 21, 29}, {103, 4, 7, 4}, {104, 3, 4, 11}, {105, 5, 9, 1}, {106, 2, 1, 11}, {107, 1, 3, 9}, {108, 3, 5, 8}, {109, 6, 9, 26}, {110, 2, 5, 1},
{111, 2, 3, 10}, {112, 1, 4, 7}, {113, 2, 5, 2}, {114, 8, 9, 59}, {115, 3, 3, 13}, {116, 3, 1, 14}, {117, 39, 29, 1513}, {118, 2, 5, 3}, {119, 7, 6, 48}, {120, 5, 9, 4},
{121, 1, 1, 11}, {122, 14, 33, 51}, {123, 3, 2, 14}, {124, 26, 19, 671}, {125, 1, 5, 1}, {126, 1, 3, 10}, {127, 1, 4, 8}, {128, 1, 5, 2}, {129, 2, 4, 9}, {130, 8, 1, 65},
{131, 5, 3, 27}, {132, 2, 3, 11}, {133, 1, 5, 3}, {134, 2, 5, 5}, {135, 6, 11, 10}, {136, 2, 2, 12}, {137, 6, 4, 37}, {138, 10, 9, 97}, {139, 3, 6, 2}, {140, 1, 5, 4},
{141, 221, 477, 47717}, {142, 3, 3, 14}, {143, 5, 8, 16}, {144, 1, 4, 9}, {145, 2, 5, 6}, {146, 11, 11, 116}, {147, 1, 3, 11}, {148, 2, 4, 10}, {149, 1, 5, 5}, {150, 32, 13, 1023},
{151, 1, 2, 12}, {152, 3, 4, 13}, {153, 5, 9, 7}, {154, 2, 1, 13}, {155, 2, 3, 12}, {156, 4, 6, 14}, {157, 7, 13, 19}, {158, 2, 5, 7}, {159, 5, 7, 21}, {160, 1, 5, 6},
{161, 2, 2, 13}, {162, 6, 9, 27}, {163, 1, 4, 10}, {164, 9, 1, 82}, {165, 3, 5, 11}, {166, 115, 43, 13222}, {167, 5, 6, 24}, {168, 4, 7, 9}, {169, 1, 1, 13}, {170, 1, 3, 12},
{171, 3, 6, 6}, {172, 12, 27, 35}, {173, 1, 5, 7}, {174, 26, 45, 605}, {175, 6, 3, 38}, {176, 1, 2, 13}, {177, 6, 8, 31}, {178, 9, 15, 58}, {179, 3, 4, 14}, {180, 2, 3, 13},
{181, 2, 1, 14}, {182, 9, 7, 80}, {183, 3, 2, 16}, {184, 1, 4, 11}, {185, 4, 6, 15}, {186, 4, 1, 21}, {187, 4, 7, 10}, {188, 1, 5, 8}, {189, 12, 21, 108}, {190, 2, 5, 9},
{191, 15, 31, 145}, {192, 2, 4, 12}, {193, 4, 5, 18}, {194, 254, 1601, 7657}, {195, 1, 3, 13}, {196, 1, 1, 14}, {197, 15, 37, 13}, {198, 6, 5, 37}, {199, 3, 6, 8}, {200, 32, 98, 328}}