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

王守恩 · 发表于 2025-12-28 16:12:10

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}}

northwolves · 发表于 2025-12-28 20:13:31

本帖最后由 northwolves 于 2025-12-28 20:14 编辑

$((4 p^4-3) (16 p^8+408 p^4+9))^2+(144 p^4-(4 p^4-3)^2)^3-(6 p (4 p^4+3))^4=0$

northwolves · 发表于 2025-12-28 20:28:15

本帖最后由 northwolves 于 2025-12-28 20:31 编辑

$44516939274744185370319^2+166584083400^3=210990377209^4$

$167478981719419401064671678188905581523359894046195633^2 + 19717429888535546483233431616377743^3 = 409269917861933664743182206^4$

王守恩 · 发表于 2025-12-29 15:04:22

$(28 p^6)^2 + (8 p^4)^3 = (6 p^3)^4$

对每个n来说：有1个解——就有无数个解。——不敢说——有问题。

对每个n来说：有通解公式就有无数个解——没问题。譬如——n=1。——就有无数个解。

$1=a^2+b^3-c^4$
$1=(k^2)^2+1^3-k^4$
$1=1^2+(k^4)^3-(k^3)^4$
$4=2^2+(k^4)^3-(k^3)^4$
$8=(k^2)^2+2^3-k^4$
$16=4^2+(k^4)^3-(k^3)^4$
$25=5^2+(k^4)^3-(k^3)^4$
$27=(k^2)^2+3^3-k^4$

对每个n来说：没有通解公式——也有无数个解？——这是重点。

王守恩 · 发表于 2025-12-30 11:45:53

题目: a,b,c=正整数。$a^x+b^y-c^z=n$ n能跑遍所有正整数吗？

能跑遍所有正整数=有解。不能跑遍所有正整数=无解。

大胆的预测一下。

6=x+y+z,
{2,2,2}=有解。
7=x+y+z,
{2,3,2}=有解。
{2,2,3}=有解。
8=x+y+z,
{2,4,2}=有解。
{2,2,4}=无解。
{3,3,2}=有解。
{2,3,3}=有解。
9=x+y+z,
{2,5,2}=有解。
{2,2,5}=有解。
{3,4,2}=有解。
{2,4,3}=无解。
{2,3,4}=有解。
{3,3,3}=无解。
10=x+y+z,
{2,6,2}=有解。
{2,2,6}=无解。
{3,5,2}=有解。
{2,5,3}=有解。
{2,3,5}=有解。
{4,4,2}=无解。
{2,4,4}=无解。
{3,4,3}=无解。
{3,3,4}=无解。
11=x+y+z,
{2,7,2}=有解。
{2,2,7}=有解。
其余的都是无解。
12=x+y+z,
{2,8,2}=有解。
其余的都是无解。
13=x+y+z,
{2,9,2}=有解。
其余的都是无解。
14=x+y+z,
{2,10,2}=有解。
其余的都是无解。

特别地，{2,y,2}应该有一个漂亮的通解公式。

补充内容 (2025-12-31 15:44):
重新复核。发现{2,4,3}=有解。

王守恩 · 发表于 2025-12-30 17:15:39

northwolves 发表于 2025-12-28 20:13
$((4 p^4-3) (16 p^8+408 p^4+9))^2+(144 p^4-(4 p^4-3)^2)^3-(6 p (4 p^4+3))^4=0$

接楼上。{2,y,2}应该有一个漂亮的通解公式。

题目: a,b,c=正整数。$a^2+b^y-c^2=n$ ——n能跑遍所有正整数。

y可以是1,2,3,4,5,6,7,......。b恰好=1或2。譬如y=9。

x=2; y=9; z=2;Table[Do[Block[{d=n + c^x}, For[b=Floor[d^(1/y)], b > 0, b--, Block[{a=(d - b^y)^(1/z)}, If[IntegerQ[a] && a > 0, Return[{n, c, b, a}]]]]], {c, Infinity}], {n, 6, 100}]

复制代码

{{6, 2, 1, 3}, {7, 53, 2, 48}, {8, 3, 1, 4}, {9, 1, 1, 3}, {10, 4, 1, 5},
{11, 85, 2, 82}, {12, 5, 1, 6}, {13, 2, 1, 4}, {14, 6, 1, 7}, {15, 39, 2, 32},
{16, 1, 1, 4}, {17, 3, 1, 5}, {18, 8, 1, 9}, {19, 23, 2, 6}, {20, 9, 1, 10},
{21, 4, 1, 6}, {22, 2, 1, 5}, {23, 83, 2, 80}, {24, 11, 1, 12}, {25, 1, 1, 5},
{26, 12, 1, 13}, {27, 51, 2, 46}, {28, 3, 1, 6}, {29, 6, 1, 8}, {30, 14, 1, 15},
{31, 25, 2, 12}, {32, 15, 1, 16}, {33, 2, 1, 6}, {34, 4, 1, 7}, {35, 31, 2, 22},
{36, 1, 1, 6}, {37, 8, 1, 10}, {38, 18, 1, 19}, {39, 27, 2, 16}, {40, 5, 1, 8},
{41, 3, 1, 7}, {42, 20, 1, 21}, {43, 37, 2, 30}, {44, 21, 1, 22}, {45, 10, 1, 12},
{46, 2, 1, 7}, {47, 23, 2, 8}, {48, 23, 1, 24}, {49, 1, 1, 7}, {50, 24, 1, 25},
{51, 231, 2, 230}, {52, 7, 1, 10}, {53, 12, 1, 14}, {54, 26, 1,27}, {55, 229, 2, 228},
{56, 3, 1, 8}, {57, 5, 1, 9}, {58, 8, 1, 11}, {59, 77, 2, 74}, {60, 29, 1, 30},
{61, 2, 1, 8}, {62, 30, 1, 31}, {63, 154, 3, 64}, {64, 1, 1, 8}, {65, 6, 1, 10},
{66, 4, 1, 9}, {67, 47, 2, 42}, {68, 33, 1, 34}, {69, 16, 1, 18}, {70, 10, 1, 13},
{71, 29, 2, 20}, {72, 21, 2, 1}, {73, 3, 1, 9}, {74, 36, 1, 37}, {75, 21, 2, 2},
{76, 5, 1, 10}, {77, 18, 1, 20}, {78, 2, 1, 9}, {79, 198, 3, 140}, {80, 21, 2, 3},
{81, 1, 1, 9}, {82, 12, 1, 15}, {83, 23, 2, 10}, {84, 41, 1, 42}, {85, 4, 1, 10},
{86, 6, 1, 11}, {87, 21, 2, 4}, {88, 13, 1, 16}, {89, 9, 1, 13}, {90, 44, 1, 45},
{91, 141, 3, 17}, {92, 3, 1, 10}, {93, 22, 1, 24}, {94, 14, 1, 17}, {95, 71, 2, 68},
{96, 7, 1, 12}, {97, 2, 1, 10}, {98, 48, 1, 49}, {99, 33, 2, 26}, {100, 1, 1, 10}}

把b=1或2固定了。

Table[FindInstance[{n == a^2 + Floor[(Cos[(n + 1) Pi/2] + 3)/2]^9 - c^2, 0 < a, 0 < c}, {a, c}, Integers, 1], {n, 6, 100}]

复制代码

{{{a -> 3, c -> 2}}, {{a -> 252, c -> 253}}, {{a -> 4, c -> 3}}, {{a -> 3, c -> 1}}, {{a -> 5, c -> 4}},
{{a -> 250, c -> 251}}, {{a -> 6, c -> 5}}, {{a -> 4, c -> 2}}, {{a -> 7, c -> 6}}, {{a -> 248, c -> 249}},
{{a -> 8, c -> 7}}, {{a -> 5, c -> 3}}, {{a -> 9, c -> 8}}, {{a -> 246, c -> 247}}, {{a -> 10, c -> 9}},
{{a -> 6, c -> 4}}, {{a -> 11, c -> 10}}, {{a -> 244, c -> 245}}, {{a -> 12, c -> 11}}, {{a -> 7, c -> 5}},
{{a -> 13, c -> 12}}, {{a -> 242, c -> 243}}, {{a -> 14, c -> 13}}, {{a -> 8, c -> 6}}, {{a -> 15, c -> 14}},
{{a -> 240, c -> 241}}, {{a -> 16, c -> 15}}, {{a -> 9, c -> 7}}, {{a -> 17, c -> 16}}, {{a -> 238, c -> 239}},
{{a -> 18, c -> 17}}, {{a -> 10, c -> 8}}, {{a -> 19, c -> 18}}, {{a -> 236, c -> 237}}, {{a -> 20, c -> 19}},
{{a -> 11, c -> 9}}, {{a -> 21, c -> 20}}, {{a -> 234, c -> 235}}, {{a -> 22, c -> 21}}, {{a -> 12, c -> 10}},
{{a -> 23, c -> 22}}, {{a -> 232, c -> 233}}, {{a -> 24, c -> 23}}, {{a -> 13, c -> 11}}, {{a -> 25, c -> 24}},
{{a -> 230, c -> 231}}, {{a -> 26, c -> 25}}, {{a -> 14, c -> 12}}, {{a -> 27, c -> 26}}, {{a -> 228, c -> 229}},
{{a -> 28, c -> 27}}, {{a -> 15, c -> 13}}, {{a -> 29, c -> 28}}, {{a -> 226, c -> 227}}, {{a -> 30, c -> 29}},
{{a -> 16, c -> 14}}, {{a -> 31, c -> 30}}, {{a -> 224, c -> 225}}, {{a -> 32, c -> 31}}, {{a -> 17, c -> 15}},
{{a -> 33, c -> 32}}, {{a -> 222, c -> 223}}, {{a -> 34, c -> 33}}, {{a -> 18, c -> 16}}, {{a -> 35, c -> 34}},
{{a -> 220, c -> 221}}, {{a -> 36, c -> 35}}, {{a -> 19, c -> 17}}, {{a -> 37, c -> 36}}, {{a -> 218, c -> 219}},
{{a -> 38, c -> 37}}, {{a -> 20, c -> 18}}, {{a -> 39, c -> 38}}, {{a -> 216, c -> 217}}, {{a -> 40, c -> 39}},
{{a -> 21, c -> 19}}, {{a -> 41, c -> 40}}, {{a -> 214, c -> 215}}, {{a -> 42, c -> 41}}, {{a -> 22,c -> 20}},
{{a -> 43, c -> 42}}, {{a -> 212, c -> 213}}, {{a -> 44, c -> 43}}, {{a -> 23, c -> 21}}, {{a -> 45, c -> 44}},
{{a -> 210, c -> 211}}, {{a -> 46, c -> 45}}, {{a -> 24, c -> 22}}, {{a -> 47, c -> 46}}, {{a -> 208, c -> 209}},
{{a -> 48, c -> 47}}, {{a -> 25, c -> 23}}, {{a -> 49, c -> 48}}, {{a -> 206, c -> 207}}, {{a -> 50, c -> 49}}}

为了看得仔细一点。特别地把y单独提出——还是很有规律的——虽然前面几行左面几列规律不明显。——当n,y够大时规律就出来了。

Table[Flatten@Table[b /. FindInstance[{n == a^2 + b^y - c^2, 0 < a, 1 <= b <= 2, 0 < c}, {b, c, a}, Integers, 1], {n, 60}], {y, 33}]

复制代码

{1, 2, b, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1},
{1, b, b, 2, b, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2},
{1, b, 2, 1, 2, 1, b, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2},
{1, b, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, b, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2},
{1, b, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, b, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1},
{1, b, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1}}

看下面的, 规律就出来了。

求助: 来个“通项公式”。谢谢！！！

王守恩 · 发表于 2025-12-31 07:15:37

2026 $\D=U_{1}^2+V_{1}^1-W_{1}^2=U_{2}^2+V_{2}^2-W_{2}^2=U_{3}^2+V_{3}^3-W_{3}^2=\cdots=U_{i}^2+V_{i}^i-W_{i}^2=\cdots=U_{2026}^2+V_{2026}^{2026}-W_{2026}^2$

说明: $U_{i},V_{i},W_{i}$ = 互不相同正整数。

王守恩 · 发表于 2026-1-1 12:38:06

补充调整。

题目: a, b, c = 正整数。$a^x + b^y - c^z$ n 能跑遍所有正整数吗？

能跑遍所有正整数 = 有解。不能跑遍所有正整数 = 无解。

大胆预测。

6=x+y+z,
{2,2,2}=有解。
7=x+y+z,
{2,3,2}=有解。
{2,2,3}=有解。
8=x+y+z,
{2,4,2}=有解。
{2,2,4}=无解。——{6, 11, 14, 22, 27, 30, 38, 43, 46, 54, 59, 62, 70, 75, 78, 86, 91, 94, 102, 107, 110, 118, 123} Mod 16={6, 11, 14}——详见19#
{3,3,2}=有解。
{2,3,3}=有解。
9=x+y+z,
{2,5,2}=有解。
{2,2,5}=有解。
{3,4,2}=有解。
{2,4,3}=有解。
{2,3,4}=有解。
{3,3,3}=无解。——A060464——Numbers that are not congruent to 4 or 5 mod 9.——4, 5, 13, 14, 22, 23, 31, 32, 40, 41, 49, 50, 58, 59, 67, 68, 76, 77, 85, 86, 94, 95,
10=x+y+z,
{2,6,2}=有解。
{2,2,6}=无解。
{3,5,2}=有解。
{2,5,3}=有解。
{2,3,5}=有解。
{4,4,2}=无解。
{2,4,4}=无解。
{3,4,3}=无解。
{3,3,4}=无解。
11=x+y+z,
{2,7,2}=有解。
{2,2,7}=有解。
其余的都是无解。
12=x+y+z,
{2,8,2}=有解。
其余的都是无解。
13=x+y+z,
{2,9,2}=有解。
{2,2,9}= ？？
其余的都是无解。
14=x+y+z,
{2,10,2}=有解。
其余的都是无解。
15=x+y+z,
{2,11,2}=有解。
其余的都是无解。
......

省略了: x, y, z 只要出现 "1", 肯定有解。

只有{2,y,2}有一个漂亮的通解公式。——见36#,37#。

王守恩 · 发表于 2026-1-2 09:42:00

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$

求助。$n=a^2+b^2-c^9$——n = 54, 92。我是怎么也鼓捣不出来。

直觉：$n=a^2+b^2-c^9$对于任意正整数n都会有解。直觉：$n=a^2+b^2-c^9$比$n=a^2+b^3-c^5$还难。

x = 2; y = 2; z = 9; Table[Do[Block[{d = n + c^z}, For[b = Floor[d^(1/y)], b ≥ 0, b--, Block[{a = (d - b^y)^(1/x)}, If[IntegerQ[a] && a ≥ 0, Return[{n, c, b, a}]]]]], {c, 32}], {n, 100}]

复制代码

{1, 1, 1, 1}, {2, 2, 17, 15}, {3, 1, 2, 0}, {4, 1, 2, 1}, {5, 5, 1397, 39},
{6, 19, 568021, 6312}, {7, 1, 2, 2}, {8, 1, 3, 0}, {9, 1, 3, 1}, {10, 2, 21, 9},
{11, 9, 19634, 1388}, {12, 1, 3, 2}, {13, 6, 3053, 870}, {14, 3, 116, 79}, {15, 1, 4, 0},
{16, 1, 4, 1}, {17, 1, 3, 3}, {18, 2, 23, 1}, {19, 1, 4, 2}, {20, 5, 1296, 523},
{21, 2, 23, 2}, {22, 7, 5227, 3610}, {23, 17, 344286, 7418}, {24, 1, 5, 0}, {25, 1, 5, 1},
{26, 2, 23, 3}, {27, 31, 4269077, 2866113}}, {28, 1, 5, 2}, {29, 2, 21, 10}, {30, 7, 6271, 1014},
{31, 1, 4, 4}, {32, 2, 20, 12}, {33, 1, 5, 3}, {34, 3, 111, 86}, {35, 1, 6, 0},
{36, 1, 6, 1}, {37, 2, 18, 15}, {38, 3, 140, 11}, {39, 1, 6, 2}, {40, 1, 5, 4},
{41, 6, 3131, 524}, {42, 2, 23, 5}, {43, 7, 6323, 611}, {44, 1, 6, 3}, {45, 2, 19, 14},
{46, 3, 127, 60}, {47, 3, 137, 31}, {48, 1, 7, 0}, {49, 1, 7, 1}, {50, 2, 21, 11},
{51, 1, 6, 4}, {52, 1, 7, 2}, {53, 2, 23, 6}, 54——Null, {55, 13, 102422, 10688},
{56, 17, 331797, 92188}, {57, 1, 7, 3}, {58, 4, 479, 181}, {59, 9, 19672, 658}, {60, 1, 6, 5},
{61, 3, 140, 12}, {62, 7, 5190, 3663}, {63, 1, 8, 0}, {64, 1, 8, 1}, {65, 2, 24, 1},
{66, 2, 23, 7}, {67, 1, 8, 2}, {68, 2, 24, 2}, {69, 5, 1385, 187}, {70, 3, 123, 68},
{71, 1, 6, 6}, {72, 1, 8, 3}, {73, 1, 7, 5}, {74, 2, 19, 15}, {75, 29, 3774250, 512038},
{76, 21, 819201, 350984}, {77, 5, 1131, 821}, {78, 7, 5782, 2631}, {79, 1, 8, 4}, {80, 1, 9, 0},
{81, 1, 9, 1}, {82, 4, 499, 115}, {83, 21, 843490, 287758}, {84, 1, 9, 2}, {85, 10, 30247, 9226},
{86, 3, 140, 13}, {87, 11, 48023, 7193}, {88, 1, 8, 5}, {89, 1, 9, 3}, {90, 3, 138, 27},
{91, 13, 102758, 6730}, 92——Null, {93, 2, 22, 11}, {94, 3, 129, 56}, {95, 5, 1368, 286},
{96, 1, 9, 4}, {97, 1, 7, 7}, {98, 2, 23, 9}, {99, 1, 10, 0}, {100, 1, 10, 1}}

王守恩 · 发表于 2026-1-2 15:34:33

题目: a, b, c = 正整数。$a^x+b^y-c^z=n$ n 能跑遍所有正整数吗？

能跑遍所有正整数 = 有解。不能跑遍所有正整数 = 无解。

我们约定 x, y, z 其中总有 1 个数 = 2,  作大胆预测。欢迎批评！

一，3个2。

1, {2,2,2}=有解。——3#, 6#, 7#。

二，2个2。
1, {2,y,2}=有解。——有一个漂亮的通解公式。——见36#,37#。
{2,3,2}=有解。
{2,4,2}=有解。
{2,5,2}=有解。
{2,6,2}=有解。
{2,7,2}=有解。
{2,8,2}=有解。
{2,9,2}=有解。

2, {2,2,z}。
{2,2,3}=有解。——11#。
{2,2,4}=无解。——{6, 11, 14, 22, 27, 30, 38, 43, 46, 54, 59, 62, 70, 75, 78, 86, 91, 94, 102, 107, 110, 118, 123} Mod 16={6, 11, 14}——详见19#
{2,2,5}=有解。
{2,2,6}=无解。
{2,2,7}=有解。
{2,2,8}=无解。
{2,2,9}=有解。

三，1个2。
1, {x,y,2}。
{3,3,2}=有解。
{3,4,2}=有解。
{3,5,2}=有解。
{4,4,2}=无解。
{3,6,2}=有解。
{4,5,2}=有解。
{3,7,2}=有解。
{4,6,2}=无解。
{5,5,2}=有解。
{3,8,2}=有解。
{4,7,2}=有解。
{5,6,2}=有解。
{3,9,2}=有解。
{4,8,2}=无解。
{5,7,2}=有解。
{6,6,2}=无解。

2, {2,y,z}。
{2,3,3}=有解。——11#。
{2,3,4}=有解。
{2,4,3}=有解。
{2,3,5}=有解。
{2,4,4}=无解。
{2,5,3}=有解。
{2,3,6}=有解。
{2,4,5}=有解。
{2,5,4}=有解。
{2,6,3}=有解。
{2,3,7}=有解。
{2,4,6}=无解。
{2,5,5}=有解。
{2,6,4}=无解。
{2,7,3}=有解。
{2,3,8}=有解。
{2,4,7}=有解。
{2,5,6}=有解。
{2,6,5}=有解。
{2,7,4}=有解。
{2,8,3}=有解。
{2,3,9}=有解。
{2,4,8}=无解。
{2,5,7}=有解。
{2,6,6}=无解。
{2,7,5}=有解。
{2,8,4}=无解。
{2,9,3}=有解。

说明。
1, 没用 "1"。 x, y, z只要有"1", 肯定有解。
2, 没用 "2"。 x, y, z至少是{3,3,3}=无解。——A060464——Numbers that are not congruent to 4 or 5 mod 9.——4, 5, 13, 14, 22, 23, 31, 32, 40, 41, 49, 50, 58, 59, 67, 68, 76, 77, 85, 86, 94, 95,
没用 "2"。太难了。想想。一个团队使用 400 万计算机小时才找到 3 为三个立方和的第三种方法。
第三种方法。3 = 569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3。
第二种方法。3 = 4^3 + 4^3 + (-5)^3。
第一种方法。3 = 1^3 + 1^3 + 1^3。
3, 对 n=a^x+b^y-c^z, 之所以选n, 我觉得n有了,n^k自然就有了。
      n=a^x+b^y-c^z 比 n=a^x+b^y-k(k是常数)还是要困难些。
      n=a^x+b^y-c^z 比 n=a^x+b^y-c^z+d^w无解的可能更大。相对 n=a^x+b^y-c^z+d^w...越长越容易找到解。

重要的事情说三遍。
$n=a^2+b^2+c^2$——n 不可以跑遍所有自然数。 $n=a^2+b^2-c^2$ ——n 可以跑遍所有自然数！
n=$a^2+b^2$数字串A,  无解。n=$a^2+b^2-c^4$数字串B,  无解。数字串B是数字串A的真子集。
数字串A——{3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 83, 84, 86, 87}——数字串A——就怎么也找不到通项公式！
数字串B——{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, 219}——数字串B——可以有通项公式。

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[投票] n^2=a^2+b^2-1

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