(a^2-1)(b^2-1)=c^3

王守恩

\((1^2-1)(2^2-1)=0^3\)

\((3^2-1)(3^2-1)=4^3\)

\((7^2-1)(17^2-1)=24^3\)

\((6^2-1)(99^2-1)=70^3\)

\((8^2-1)(97^2-1)=84^3\)

\((97^2-1)(251^2-1)=840^3\)

\((199^2-1)(241^2-1)=1320^3\)

\((151^2-1)(721^2-1)=2280^3\)

\((129^2-1)(1351^2-1)=3120^3\)

\((25^2-1)(24335^2-1)=7176^3\)

4 = 2^2

24 = 2^3 * 3

70 = 2 * 5 * 7

84 = 2^2 * 3 * 7

840 = 2^3 * 3 * 5 * 7

1320 = 2^3 * 3 * 5 * 11

2280 = 2^3 * 3 * 5 * 19

3120 = 2^4 * 3 * 5 * 13

7176 = 2^3 * 3 * 13 * 23

{0, 4, 24, 70, 84, 840, 1320, 2280, 3120, 7176}——OEIS没有。

再来几个——我也算不了。——求助！！！

风花飘飘

没有了，就这么多。

风花飘飘

（1^2-1）(b^2-1)≡0^3————A

（a^2-1）(1^2-1)≡0^3————B

风花飘飘

哈哈，来就来……
把a,b都统统加上负号！

这题满足条件的解：
目前总共有44组，
其它的我暂时找不到！

王守恩

{0, 4, 24, 70, 84, 840, 1320, 2280, 3120, 7176}——共10个。

{1351, 13311, 68640}——第11个还是有的。

{0, 4, 24, 70, 84, 840, 1320, 2280, 3120, 7176, 68640}——共11个。

王守恩

题目有点难——自己给自己打打气！

关键还是我们太超前了！OEIS 连这个也没有！

\((1^2-1)(2^2-1)=0^2\)
\((2^2-1)(7^2-1)=12^2\)
\((2^2-1)(26^2-1)=45^2\)
\((3^2-1)(17^2-1)=48^2\)
\((4^2-1)(31^2-1)=120^2\)
\((2^2-1)(97^2-1)=168^2\)
\((7^2-1)(26^2-1)=180^2\)
\((5^2-1)(49^2-1)=240^2\)
\((3^2-1)(99^2-1)=280^2\)
\((6^2-1)(71^2-1)=420^2\)
\((2^2-1)(362^2-1)=627^2\)
\((7^2-1)(97^2-1)=672^2\)
\((4^2-1)(244^2-1)=945^2\)

{0, 12, 45, 48, 120, 168, 180, 240, 280, 420, 627, 672, 945, 1008, 1440, 1632, 1680, 1980, 2376, 2508, 2520, 2640, 3432, 4368, 5005, 5460, 6720, 7560, 8160, 9405, 9792, 11628, 13680, 15960, 18480, 21252, 23760, 35112}

OEIS没有这串数。——再简单一点——OEIS还是可以有的。

\((2^2-1)(7^2-1)=12^2\)
\((2^2-1)(26^2-1)=45^2\)
\((2^2-1)(97^2-1)=168^2\)
\((2^2-1)(362^2-1)=627^2\)
\((2^2-1)(1351^2-1)=2340^2\)
\((2^2-1)(5042^2-1)=8733^2\)
\((2^2-1)(18817^2-1)=32592^2\)
\((2^2-1)(70266^2-1)=121635^2\)
\((2^2-1)(262087^2-1)=453948^2\)
\((2^2-1)(978122^2-1)=1694157^2\)

A001075——7, 26, 97, 362, 1351, 5042, 18817, 70226, 262087, 978122, 3650401, 13623482, 50843527, 189750626, 708158977, 2642885282, 9863382151, 36810643322, 137379191137, 512706121226, 1913445293767,
A005320——12, 45, 168, 627, 2340, 8733, 32592, 121635, 453948, 1694157, 6322680, 23596563, 88063572, 328657725, 1226567328, 4577611587, 17083879020, 63757904493, 237947738952, 888033051315, 3314184466308,  

王守恩

谢谢 northwolves ！——每个 n 取第 1 个。

\((1^2-1)(1^2-1)=0^2\)
\((2^2-1)(7^2-1)=12^2\)
\((3^2-1)(17^2-1)=48^2\)
\((4^2-1)(31^2-1)=120^2\)
\((5^2-1)(49^2-1)=240^2\)
\((6^2-1)(71^2-1)=420^2\)
\((7^2-1)(97^2-1)=672^2\)
\((8^2-1)(127^2-1)=1008^2\)
\((9^2-1)(161^2-1)=1440^2\)

{{1, 1, 0}, {2, 7, 12}, {3, 17, 48}, {4, 31, 120}, {5, 49, 240}, {6, 71, 420}, {7, 97, 672}, {8, 127, 1008}, {9, 161, 1440}, {10, 199, 1980}, {11, 241, 2640},
{12, 287, 3432}, {13, 337, 4368}, {14, 391, 5460}, {15, 449, 6720}, {16, 511, 8160}, {17, 577, 9792}, {18, 647, 11628}, {19, 721, 13680}, {20, 799, 15960}}

Table[{n, 2 n^2 - 1, 2 n (n^2 - 1)}, {n, 20}]

王守恩

搬运一下。
组合(1)。a = 3; b = 3; c = 4;
a^2-1 = {{2,3}}
b^2-1 = {{2,3}}
c^3    = {{2,6}}
组合(2)。a = 6; b = 99; c = 70;
a^2-1 = {{5,1},{7,1}}
b^2-1 = {{2,3},{5,2},{7,2}}
c^3    = {{2,3},{5,3},{7,3}}
组合(3)。a = 7; b = 17; c = 24;
a^2-1 = {{2,4},{3,1}}
b^2-1 = {{2,5},{3,2}}
c^3    = {{2,9},{3,3}}
组合(4)。a = 8; b = 97; c = 84;
a^2-1 = {{3,2},{7,1}}
b^2-1 = {{2,6},{3,1},{7,2}}
c^3    = {{2,6},{3,3},{7,3}}
组合(5)。a = 25; b = 24335; c = 7176;
a^2-1 = {{2,4},{3,1},{13,1}}
b^2-1 = {{2,5},{3,2},{13,2},{23,3}}
c^3    = {{2,9},{3,3},{13,3},{23,3}}
组合(6)。a = 97; b = 251; c = 840;
a^2-1 = {{2,6},{3,1},{7,2}}
b^2-1 = {{2,3},{3,2},{5,3},{7,1}}
c^3    = {{2,9},{3,3},{5,3},{7,3}}
组合(7)。a = 129; b = 1351; c = 3120;
a^2-1 = {{2,8},{5,1},{13,1}}
b^2-1 = {{2,4},{3,3},{5,2},{13,2}}
c^3    = {{2,12},{3,3},{5,3},{13,3}}
组合(8)。a = 151; b = 721; c = 2280;
a^2-1 = {{2,4},{3,1},{5,2},{19,1}}
b^2-1 = {{2,5},{3,2},{5,1},{19,2}}
c^3    = {{2,9},{3,3},{5,3},{19,3}}
组合(9)。a = 199; b = 241; c = 1320;
a^2-1 = {{2,4},{3,2},{5,2},{11,1}}
b^2-1 = {{2,5},{3,1},{5,1},{11,2}}
c^3    = {{2,9},{3,3},{5,3},{11,3}}
组合(10)。a = 1351; b = 13311; c = 68640;
a^2-1 = {{2,4},{3,3},{5,2},{13,2}}
b^2-1 = {{2,11},{5,1},{11,3},{13,1}}
c^3    = {{2,15},{3,3},{5,3},{11,3},{13,3}}
组合(11)。a = 10405; b = 3786751; c = 11579040;——谢谢网友 northwolves ！
a^2-1 = {{2,3},{3,2},{11,2},{17,2},{43,1}}
b^2-1 = {{2,12},{3,4},{5,3},{11,1},{17,1},{43,2}}
c^3    = {{2,15},{3,6},{5,3},{11,3},{17,3},{43,3}}
组合(12)。a = 708158977; b = 3730418047; c = 1910991897024;——谢谢网友 nindilg ！
a^2-1 = {{2,10},{3,1},{7,2},{31,2},{97,2},{607,2}}
b^2-1 = {{2,8},{3,2},{7,1},{19,3},{31,1},{41,3},{97,1},{607,1}}
c^3    = {{2,18},{3,3},{7,3},{19,3},{31,3},{41,3},{97,3},{607,3}}

王守恩

k = 288; W[p_] := Log[k]/Log[p]; U = {}; Do[V = 2^a*3^b; If[V <= k, n = Sqrt[V + 1];
If[IntegerQ[n], AppendTo[U, {a, b, n}]]], {a, 0, W[2]}, {b, W[3]}]; SortBy[U, Last]
{{0, 1, 2}, {3, 1, 5}, {4, 1, 7}, {5, 2, 17}}
\(n^2-1=(n-1)(n+1)=2^a*3^b, b > 0\)——这里 k = 288 有 4 个解。问题来了——不管如何加大 k, 还是这 4 个解。为什么呀？

k = 25920; W[p_] := Log[k]/Log[p]; U = {}; Do[V = 2^a*3^b*5^c; If[V <= k, n = Sqrt[V + 1];
If[IntegerQ[n], AppendTo[U, {a, b, c, n}]]], {a, 0, W[2]}, {b, 0, W[3]}, {c, W[5]}]; SortBy[U, Last]
{{0, 1, 1, 4}, {4, 0, 1, 9}, {3, 1, 1, 11}, {3, 2, 1, 19}, {0, 3, 2, 26}, {6, 1, 1, 31}, {5, 1, 2, 49}, {6, 4, 1, 161}}
\(n^2-1=(n-1)(n+1)=2^a*3^b*5^c, c > 0\)——这里 k = 25920 有 8 个解。问题来了——不管如何加大 k, 还是这 8 个解。为什么呀？

k = 76545000; W[p_] := Log[k]/Log[p]; U = {}; Do[V = 2^a*3^b*5^c*7^d; If[V <= k, n = Sqrt[V + 1];
If[IntegerQ[n], AppendTo[U, {a, b, c, d, n}]]], {a, 0, W[2]}, {b, 0, W[3]}, {c, 0, W[5]}, {d, W[7]}]; SortBy[U, Last]
{{0, 0, 1, 1, 6}, {0, 2, 0, 1, 8}, {3, 1, 0, 1, 13}, {5, 0, 0, 1, 15}, {3, 1, 1, 1, 29}, {4, 1, 1, 1, 41}, {4, 3, 0, 1, 55}, {4, 2, 1, 1, 71},
{6, 1, 0, 2, 97}, {3, 0, 2, 2, 99}, {8, 2, 0, 1, 127}, {0, 5, 1, 2, 244}, {3, 2, 3, 1, 251}, {7, 2, 2, 1, 449}, {7, 1, 2, 4, 4801}, {3, 7, 4, 1, 8749}}
\(n^2-1=(n-1)(n+1)=2^a*3^b*5^c*7^d, d > 0\)——这里 k = 76545000 有 16 个解。问题来了——不管如何加大 k, 还是这 16 个解。为什么呀？

k = 384199200; W[p_] := Floor[Log[k]/Log[p]]; U = {}; Do[V = 2^a*3^b*5^c*7^d*11^f; If[V <= k, n = Sqrt[V + 1];
If[IntegerQ[n], AppendTo[U, {a, b, c, d, f, n}]]], {a, 0, W[2]}, {b, 0, W[3]}, {c, 0, W[5]}, {d, 0, W[7]}, {f, W[11]}]; SortBy[U, Last]
{{0, 2, 0, 0, 1, 10}, {3, 0, 1, 0, 1, 21}, {4, 1, 0, 0, 1, 23}, {0, 1, 1, 1, 1, 34}, {3, 1, 0, 1, 1, 43}, {7, 1, 0, 0, 1, 65}, {0, 1, 2, 1, 1, 76}, {4, 2, 1, 0, 1, 89}, {3, 3, 1, 0, 1, 109}, {5, 0, 1, 1, 1, 111},
{3, 2, 0, 2, 1, 197}, {4, 2, 2, 0, 1, 199}, {5, 1, 1, 0, 2, 241}, {6, 0, 2, 1, 1, 351}, {3, 5, 0, 0, 2, 485}, {9, 1, 1, 1, 1, 769}, {5, 2, 1, 2, 1, 881}, {4, 3, 1, 2, 1, 1079}, {6, 3, 2, 1, 2, 6049}, {5, 4, 2, 2, 2, 19601}}
\(n^2-1=(n-1)(n+1)=2^a*3^b*5^c*7^d*11^F, F > 0\)——这里 k = 384199200 有 20 个解。问题来了——不管如何加大 k, 还是这 20 个解。为什么呀？

k = 999999999999999; W[p_] := Floor[Log[k]/Log[p]]; U = {}; Do[V = 2^a*3^b*5^c*7^d*11^f*13^g; If[V <= k, n = Sqrt[V + 1];
If[IntegerQ[n], AppendTo[U, {a, b, c, d, f, g, n}]]], {a, 0, W[2]}, {b, 0, W[3]}, {c, 0, W[5]}, {d, 0, W[7]}, {f, 0, W[11]}, {g, W[13]}]; SortBy[U, Last]
{{0, 0, 0, 0, 1, 1, 12}, {0, 1, 1, 0, 0, 1, 14}, {4, 1, 0, 0, 0, 1, 25}, {3, 0, 0, 1, 0, 1, 27}, {3, 0, 2, 0, 0, 1, 51}, {3, 3, 0, 0, 0, 1, 53}, {0, 2, 1, 1, 0, 1, 64}, {5, 1, 1, 0, 0, 1, 79}, {8, 0, 1, 0, 0, 1, 129},
{3, 1, 1, 0, 1, 1, 131}, {3, 1, 0, 1, 1, 1, 155}, {3, 2, 1, 1, 0, 1, 181}, {5, 1, 1, 1, 0, 1, 209}, {0, 1, 2, 1, 1, 1, 274}, {6, 2, 0, 0, 1, 1, 287}, {5, 1, 0, 1, 0, 2, 337}, {4, 1, 1, 2, 0, 1, 391}, {4, 4, 2, 0, 0, 1, 649},
{3, 3, 2, 1, 0, 1, 701}, {7, 3, 0, 0, 1, 1, 703}, {4, 1, 0, 1, 2, 1, 727}, {0, 0, 1, 1, 2, 2, 846}, {6, 1, 4, 0, 0, 1, 1249}, {4, 3, 2, 0, 0, 2, 1351}, {5, 6, 0, 1, 0, 1, 1457}, {0, 2, 2, 1, 2, 1, 1574}, {5, 0, 3, 1, 1, 1, 2001},
{4, 1, 1, 3, 1, 1, 3431}, {7, 3, 1, 1, 1, 1, 4159}, {14, 2, 1, 1, 0, 1, 8191}, {9, 1, 2, 0, 1, 2, 8449}, {11, 0, 1, 0, 3, 1, 13311}, {5, 2, 0, 1, 3, 2, 21295}, {8, 6, 2, 1, 1, 2, 246401}}
\(n^2-1=(n-1)(n+1)=2^a*3^b*5^c*7^d*11^F*13^g, g > 0\)——这里 k = 999999999999999(?) 有 34 个解。问题来了——不管如何加大 k, 还是这 34 个解。为什么呀？

k = 999999999999999; W[p_] := Floor[Log[k]/Log[p]]; U = {}; Do[V = 2^a*3^b*5^c*7^d*11^f*13^g*17^j; If[V <= k, n = Sqrt[V + 1];
If[IntegerQ[n], AppendTo[U, {a, b, c, d, f, g, j, n}]]], {a, 0, W[2]}, {b, 0, W[3]}, {c, 0, W[5]}, {d, 0, W[7]}, {f, 0, W[11]}, {g, 0, W[13]}, {j, W[17]}]; SortBy[U, Last]
{{0, 1, 1, 0, 0, 0, 1, 16}, {6, 0, 0, 0, 0, 0, 1, 33}, {3, 2, 0, 0, 0, 0, 1, 35}, {0, 1, 0, 2, 0, 0, 1, 50}, {3, 1, 0, 0, 1, 0, 1, 67}, {3, 0, 1, 1, 0, 0, 1, 69}, {3, 1, 2, 0, 0, 0, 1, 101}, {4, 1, 0, 0, 0, 1, 1, 103},
{0, 2, 0, 1, 0, 1, 1, 118}, {0, 0, 0, 1, 2, 0, 1, 120}, {4, 1, 1, 1, 0, 0, 1, 169}, {0, 3, 0, 1, 1, 0, 1, 188}, {5, 1, 1, 1, 0, 0, 1, 239}, {5, 3, 1, 0, 0, 0, 1, 271}, {3, 2, 0, 1, 1, 0, 1, 307}, {3, 0, 1, 0, 0, 2, 1, 339},
{4, 0, 1, 0, 1, 1, 1, 441}, {10, 1, 1, 0, 0, 0, 1, 511}, {6, 1, 0, 1, 0, 1, 1, 545}, {7, 2, 0, 0, 0, 0, 2, 577}, {3, 1, 3, 0, 1, 0, 1, 749}, {3, 2, 0, 2, 0, 1, 1, 883}, {6, 1, 1, 1, 1, 0, 1, 1121}, {3, 3, 1, 1, 1, 0, 1, 1189},
{0, 4, 3, 0, 1, 0, 1, 1376}, {3, 1, 1, 1, 1, 1, 1, 1429}, {8, 0, 0, 2, 0, 1, 1, 1665}, {5, 2, 1, 0, 1, 1, 1, 1871}, {0, 4, 2, 1, 0, 0, 2, 2024}, {8, 2, 0, 0, 2, 0, 1, 2177}, {4, 1, 1, 1, 1, 0, 2, 2311}, {5, 2, 2, 2, 0, 0, 1, 2449},
{3, 1, 2, 2, 0, 1, 1, 2549}, {4, 5, 2, 1, 0, 0, 1, 3401}, {3, 1, 0, 3, 2, 0, 1, 4115}, {3, 5, 1, 0, 1, 1, 1, 4861}, {4, 1, 4, 2, 0, 0, 1, 4999}, {5, 2, 2, 0, 0, 1, 2, 5201}, {3, 3, 0, 1, 0, 1, 3, 9827}, {5, 6, 0, 3, 0, 0, 1, 11663},
{5, 2, 3, 1, 1, 1, 1, 24751}, {8, 2, 2, 1, 2, 0, 1, 28799}, {6, 1, 1, 1, 0, 4, 1, 57121}, {4, 3, 0, 4, 0, 1, 2, 62425}, {4, 7, 1, 0, 1, 2, 1, 74359}, {6, 4, 1, 4, 1, 1, 1, 388961}, {4, 2, 1, 5, 0, 3, 1, 672281}}
\(n^2-1=(n-1)(n+1)=2^a*3^b*5^c*7^d*11^F*13^g*17^j, j > 0\)——这里 k = 999999999999999(?) 有 47(?) 个解。问题来了——不管如何加大 k, 还是这 47(?) 个解。为什么呀？

王守恩

组合(1)。\((3^2-1)(3^2-1)=4^3\)
组合(2)。\((7^2-1)(17^2-1)=24^3\)
组合(3)。\((6^2-1)(99^2-1)=70^3\)
组合(4)。\((8^2-1)(97^2-1)=84^3\)
组合(5)。\((97^2-1)(251^2-1)=840^3\)
组合(6)。\((199^2-1)(241^2-1)=1320^3\)
组合(7)。\((151^2-1)(721^2-1)=2280^3\)
组合(8)。\((129^2-1)(1351^2-1)=3120^3\)
组合(9)。\((25^2-1)(24335^2-1)=7176^3\)
组合(10)。\((1351^2-1)(13311^2-1)=68640^3\)
组合(11)。\((10405^2-1)(3786751^2-1)=11579040^3\)
组合(12)。\((708158977^2-1)(3730418047^2-1)=1910991897024^3\)

\((a^2-1)(b^2-1)=c^3\)  是最困难的。\((a^2-1)(b^2-1)(c^2-1)=v^3\)  就容易些。

\((2^2-1)(2^2-1)(5^2-1)=6^3\)
\((2^2-1)(5^2-1)(5^2-1)=12^3\)
\((4^2-1)(4^2-1)(11^2-1)=30^3\)
\((3^2-1)(7^2-1)(17^2-1)=48^3\)
\((4^2-1)(4^2-1)(31^2-1)=60^3\)
\((2^2-1)(19^2-1)(26^2-1)=90^3\)
\((4^2-1)(7^2-1)(49^2-1)=120^3\)
\((3^2-1)(6^2-1)(99^2-1)=140^3\)
\((2^2-1)(13^2-1)(97^2-1)=168^3\)
\((5^2-1)(19^2-1)(26^2-1)=180^3\)

{{2, 2, 5, 6}, {2, 5, 5, 12}, {4, 4, 11, 30}, {3, 7, 17, 48}, {4, 4, 31, 60}, {2, 19, 26, 90}, {4, 7, 49, 120}, {3, 6, 99, 140}, {2, 13, 97, 168}, {4, 4, 161, 180}, {4, 8, 99, 210}, {4, 31, 31, 240}, {4, 6, 244, 315}, {5, 13, 97, 336}, {4, 11, 161, 360},
{8, 11, 99, 420}, {7, 31, 49, 480}, {9, 15, 99, 560}, {6, 11, 244, 630}, {4, 31, 161, 720}, {8, 31, 99, 840}, {23, 23, 65, 1056}, {6, 31, 244, 1260}, {21, 21, 109, 1320}, {7, 49, 161, 1440}, {3, 97, 251, 1680}, {13, 26, 244, 1890}, {13, 55, 127, 2016},
SortBy[DeleteDuplicatesBy[Reap[Do[If[a != c && IntegerQ[v = ((a^2 - 1) (b^2 - 1) (c^2 - 1))^(1/3)], Sow[{a, b, c, v}]], {a, 2, n = 99}, {b, a, n}, {c, b, n}]][[2, 1]], Last], Last]

{6, 12, 30, 48, 60, 90, 120, 140, 168, 180, 210, 240, 315, 336, 360, 420, 480, 560, 630, 720, 840, 1056, 1260, 1320, 1440, 1680, 1890, 2016, 2100, 2112, 2160, 2310, 2520, 2640, 2880, 3360, 3780, 3960, 4200, 4320, 4620, 6300, 7560, 8190}
Sort@DeleteDuplicates@Reap[Do[If[a != c && IntegerQ[v = ((a^2 - 1) (b^2 - 1) (c^2 - 1))^(1/3)], Sow[v]], {a, 2, n = 99}, {b, a, n}, {c, b, n}]][[2, 1]]