伟大的猜想

王守恩

\(x^n+(x+1)^n+(x+2)^n+\cdots+(x+j)^n=y^n+(y+1)^n+(y+2)^n+\cdots+(y+k)^n\)

x,  y,  j,  k,  n = 正整数。n > 3。如何证明无解？—— 简单！ 来个反例就行！！

风花飘飘

疯了吧？老王这是想干什么⁉

王守恩

风花飘飘 发表于 2026-8-18 10:35
疯了吧？老王这是想干什么⁉

伟大的猜想——不存在连续整数的n次方和等于连续整数的n次方和。

参考——【不服来战】总是存在n^3个连续整数的立方和等于一个立方数——谢谢网友 elim！谢谢网友 风花飘飘！

\((x+1)^3+\cdots+(x+n^3)^3=\big(\frac{(n-1)n(n+1)(n^2+2)}{6}\big)^3,\ \ 其中\  x=\frac{n^4-3n^3-2n^2-2}{6}\)

王守恩

\(x^n+(x+1)^n+(x+2)^n+\cdots+(x+j)^n=y^n+(y+1)^n+(y+2)^n+\cdots+(y+k)^n\)

x,  y,  j,  k,  n = 正整数。n > 3。如何证明无解？—— 简单！ 来个反例就行！！

王守恩

谢谢数论爱好者 ！！！

大胆猜想(1)。\(X^2=(a^n-1)(b^n-1)\)——n > 4 无解。
ParallelTable[Catch[Do[W = (a^n - 1) (b^n - 1); x = Round[Sqrt[W]]; If[x^2 == W, Throw[{n, x, a, b}]], {a, 2, 5000}, {b, a + 1, 5000}]; "未找到"], {n, 5, 9}]
{{2, 2, 5}, {12, 2, 7}, {21, 2, 4}, {9653280, 13, 239}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"}

大胆猜想(2)。\(X^2=(a^n+1)(b^n+1)\)——n > 3 无解。
ParallelTable[Catch[Do[W = (a^n + 1) (b^n + 1); x = Round[Sqrt[W]]; If[x^2 == W, Throw[{n, x, a, b}]], {a, 2, 5000}, {b, a + 1, 5000}]; "未找到"], {n, 5, 9}]
{{6, 2, 11}, {85, 2, 38}, {1953, 6, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"}

漂亮数字串(1)。\(X^2=(a^2-1)(b^2-1)\)————OEIS没有。
X = 35000; Select[Union@Cases[Flatten[Table[{Sqrt[(a^2 - 1) (b^2 - 1)], a, b}, {a, 2, Sqrt[X]}, {b, a + 1, Sqrt[X^2/3]}], 1], {_Integer, _, _}][[All, 1]], # ≤ X &]——代码还能短吗？谢谢！
{12, 45, 48, 120, 168, 180, 240, 280, 420, 627, 672, 945, 1008, 1440, 1632, 1680, 1980, 2340, 2376, 2508, 2520, 2640, 3432, 4368, 5005, 5460, 6720, 7440, 7560}

漂亮数字串(2)。\(X^2=(a^n-0)(b^n-0)\)————OEIS没有。1 < a < b
ParallelTable[Catch[Do[W = (a^n - 0) (b^n - 0); x = Round[Sqrt[W]]; If[x^2 == W, Throw[{n, x, a, b}]], {a, 2, 5000}, {b, a + 1, 5000}]; "未找到"], {n, 5, 9}]
{1, 4, 6, 64, 36, 1024, 216, 16384, 1296, 262144, 7776, 4194304, 46656, 67108864, 279936, 1073741824, 1679616, 17179869184, 10077696, 274877906944}
Table[If[EvenQ[n], 6^(n/2), 4^n], {n, 0, 19}]——通项公式可以这样。——代码还能短吗？谢谢！

王守恩

这些都是干货(不是水货)——我们能说看不懂吗？！？！？！

ParallelTable[Catch[Do[If[IntegerQ[Sqrt[(a^n - k) (b^n - k)]], Throw[{Sqrt[(a^n - k) (b^n - k)], a, b}]], {a, 2, 100}, {b, a + 1, 100}]; "未找到"], {k, -9, 9}, {n, 12}]
{{{22, 2, 35}, {195, 2, 54}, {90, 3, 6}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{20, 2, 32}, {36, 2, 10}, {1248, 2, 46}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 9}, {44, 2, 13}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到",  "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 12}, {70, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{14, 2, 23}, {42, 3, 11}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 20}, {40, 2, 14}, {258, 5, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{10, 2, 17}, {14, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 7}, {54, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 11}, {85, 2, 38}, {1953, 6, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{4, 2, 8}, {6, 2, 3}, {64, 2, 8}, {36, 2, 3}, {1024, 2, 8}, {216, 2, 3}, {16384, 2, 8}, {1296, 2, 3}, {262144, 2, 8}, {7776, 2, 3}, {4194304, 2, 8}, { 46656, 2, 3}},
{{2, 2, 5}, {12, 2, 7}, {21, 2, 4}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {14, 2, 10}, {7874, 4, 100}, "未找到", "未找到", {7874, 2, 10}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {66, 3, 27}, {132, 3, 9}, {923, 2, 16}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 4}, {0, 2, 3}, {22, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 5}, {22, 4, 7}, {39, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{2, 2, 5}, {15, 3, 9}, {74, 2, 14}, {50, 2, 4}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 7}, {6, 3, 5}, {181, 2, 32}, "未找到", {905, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 8}, {56, 4, 20}, {0, 2, 3}, {1912, 2, 26}, "未找到",  "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 9}, {0, 2, 3}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"}}

ParallelTable[Catch[Do[If[IntegerQ[Sqrt[(a^n - k) (b^n - k)]], Throw[{Sqrt[(a^n - k) (b^n - k)], a, b}]], {a, 2, 500}, {b, a + 1, 500}]; "未找到"], {k, -9, 9}, {n, 12}]
{{{22, 2, 35}, {195, 2, 54}, {90, 3, 6}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{20, 2, 32}, {36, 2, 10}, {1248, 2, 46}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 9}, {44, 2, 13}, {11955, 2, 212}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 12}, {70, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{14, 2, 23}, {42, 3, 11}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 20}, {40, 2, 14}, {4668, 2, 122}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{10, 2, 17}, {14, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 7}, {54, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 11}, {85, 2, 38}, {1953, 6, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{4, 2, 8}, {6, 2, 3}, {64, 2, 8}, {36, 2, 3}, {1024, 2, 8}, {216, 2, 3}, {16384, 2, 8}, {1296, 2, 3}, {262144, 2, 8}, {7776, 2, 3}, {4194304, 2, 8}, { 46656, 2, 3}},
{{2, 2, 5}, {12, 2, 7}, {21, 2, 4}, {9653280, 13, 239}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {14, 2, 10}, {7874, 4, 100}, "未找到", "未找到", {7874, 2, 10}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {66, 3, 27}, {132, 3, 9}, {923, 2, 16}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 4}, {0, 2, 3}, {22, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 5}, {22, 4, 7}, {39, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{2, 2, 5}, {15, 3, 9}, {74, 2, 14}, {50, 2, 4}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 7}, {6, 3, 5}, {181, 2, 32}, {3282174, 7, 259}, {905, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 8}, {56, 4, 20}, {0, 2, 3}, {1912, 2, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 9}, {0, 2, 3}, {1270980, 24, 489}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"}}

Table[Catch[Do[If[IntegerQ[Sqrt[(a^n - k) (b^n - k)]], Throw[{Sqrt[(a^n - k) (b^n - k)], a, b}]], {a, 2, 1000}, {b, a + 1, 1000}]; "未找到"], {k, -9, 9}, {n, 12}]
{{{22, 2, 35}, {195, 2, 54}, {90, 3, 6}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{20, 2, 32}, {36, 2, 10}, {1248, 2, 46}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 9}, {44, 2, 13}, {11955, 2, 212}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 12}, {70, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{14, 2, 23}, {42, 3, 11}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 20}, {40, 2, 14}, {4668, 2, 122}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{10, 2, 17}, {14, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 7}, {54, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 11}, {85, 2, 38}, {1953, 6, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{4, 2, 8}, {6, 2, 3}, {64, 2, 8}, {36, 2, 3}, {1024, 2, 8}, {216, 2, 3}, {16384, 2, 8}, {1296, 2, 3}, {262144, 2, 8}, {7776, 2, 3}, {4194304, 2, 8}, { 46656, 2, 3}},
{{2, 2, 5}, {12, 2, 7}, {21, 2, 4}, {9653280, 13, 239}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {14, 2, 10}, {7874, 4, 100}, "未找到", "未找到", {7874, 2, 10}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {66, 3, 27}, {132, 3, 9}, {923, 2, 16}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 4}, {0, 2, 3}, {22, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 5}, {22, 4, 7}, {39, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{2, 2, 5}, {15, 3, 9}, {74, 2, 14}, {50, 2, 4}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 7}, {6, 3, 5}, {181, 2, 32}, {3282174, 7, 259}, {905, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 8}, {56, 4, 20}, {0, 2, 3}, {1912, 2, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 9}, {0, 2, 3}, {1270980, 24, 489}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"}}

Table[Catch[Do[If[IntegerQ[Sqrt[(a^n - k) (b^n - k)]], Throw[{Sqrt[(a^n - k) (b^n - k)], a, b}]], {a, 2, 2000}, {b, a + 1, 2000}]; "未找到"], {k, -9, 9}, {n, 12}]
{{{22, 2, 35}, {195, 2, 54}, {90, 3, 6}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{20, 2, 32}, {36, 2, 10}, {1248, 2, 46}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 9}, {44, 2, 13}, {11955, 2, 212}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 12}, {70, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{14, 2, 23}, {42, 3, 11}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 20}, {40, 2, 14}, {4668, 2, 122}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{10, 2, 17}, {14, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 7}, {54, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 11}, {85, 2, 38}, {1953, 6, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{4, 2, 8}, {6, 2, 3}, {64, 2, 8}, {36, 2, 3}, {1024, 2, 8}, {216, 2, 3}, {16384, 2, 8}, {1296, 2, 3}, {262144, 2, 8}, {7776, 2, 3}, {4194304, 2, 8}, { 46656, 2, 3}},
{{2, 2, 5}, {12, 2, 7}, {21, 2, 4}, {9653280, 13, 239}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {14, 2, 10}, {7874, 4, 100}, "未找到", "未找到", {7874, 2, 10}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {66, 3, 27}, {132, 3, 9}, {923, 2, 16}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 4}, {0, 2, 3}, {22, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 5}, {22, 4, 7}, {39, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{2, 2, 5}, {15, 3, 9}, {74, 2, 14}, {50, 2, 4}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 7}, {6, 3, 5}, {181, 2, 32}, {3282174, 7, 259}, {905, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 8}, {56, 4, 20}, {0, 2, 3}, {1912, 2, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 9}, {0, 2, 3}, {1270980, 24, 489}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"}}

Table[Catch[Do[If[IntegerQ[Sqrt[(a^n - k) (b^n - k)]], Throw[{Sqrt[(a^n - k) (b^n - k)], a, b}]], {a, 2, 5000}, {b, a + 1, 5000}]; "未找到"], {k, -9, 9}, {n, 12}]
{{{22, 2, 35}, {195, 2, 54}, {90, 3, 6}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{20, 2, 32}, {36, 2, 10}, {1248, 2, 46}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 9}, {44, 2, 13}, {11955, 2, 212}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 12}, {70, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{14, 2, 23}, {42, 3, 11}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{12, 2, 20}, {40, 2, 14}, {4668, 2, 122}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{10, 2, 17}, {14, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 7}, {54, 2, 22}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{6, 2, 11}, {85, 2, 38}, {1953, 6, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{4, 2, 8}, {6, 2, 3}, {64, 2, 8}, {36, 2, 3}, {1024, 2, 8}, {216, 2, 3}, {16384, 2, 8}, {1296, 2, 3}, {262144, 2, 8}, {7776, 2, 3}, {4194304, 2, 8}, { 46656, 2, 3}},
{{2, 2, 5}, {12, 2, 7}, {21, 2, 4}, {9653280, 13, 239}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {14, 2, 10}, {7874, 4, 100}, "未找到", "未找到", {7874, 2, 10}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 3}, {66, 3, 27}, {132, 3, 9}, {923, 2, 16}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 4}, {0, 2, 3}, {22, 2, 5}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 5}, {22, 4, 7}, {39, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{2, 2, 5}, {15, 3, 9}, {74, 2, 14}, {50, 2, 4}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 7}, {6, 3, 5}, {181, 2, 32}, {3282174, 7, 259}, {905, 2, 8}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 8}, {56, 4, 20}, {0, 2, 3}, {1912, 2, 26}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"},
{{0, 2, 9}, {0, 2, 3}, {1270980, 24, 489}, "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到", "未找到"}}

王守恩

n 个连续数的积 + n 个连续数的积 = n 个连续数的积。——还真找了一个！！！——还是无限的！！！

\(9×10×11×12×13×14+10×11×12×13×14×15=11×12×13×14×15×16\)

\(64×65×66×67×\cdots×103+65×66×67×68×\cdots×104=66×67×68×69×\cdots×105\)

\(441×442×443×444×\cdots×713+442×443×444×445×\cdots×714=443×444×445×446×\cdots×715\)

\(3025×3026×3027×3028×\cdots×4894+3026×3027×3028×3029×\cdots×4895=3027×3028×3029×3030×\cdots×4896\)

\(20736×20737×20738×20739×\cdots×33551+20737×20738×20739×20740×\cdots×33552=20738×20739×20740×20741×\cdots×33553\)

\(142129×142130×142131×142132×\cdots×229969+142130×142131×142132×142133×\cdots×229970=142131×142132×142133×142134×\cdots×229971\)

A={10, 65, 442, 3026, 20737, 142130, 974170, 6677057, 45765226, 313679522, 2149991425, 14736260450, 101003831722, 692290561601, 4745030099482, 32522920134770, 222915410843905, 1527884955772562}
A的代码——Table[Fibonacci[2 n + 1] Fibonacci[2 n + 3], {n, 9}]

B={15, 104, 714, 4895, 33552, 229970, 1576239, 10803704, 74049690, 507544127, 3478759200, 23843770274, 163427632719, 1120149658760, 7677619978602, 52623190191455, 360684711361584, 2472169789339634}
B的代码——Table[Fibonacci[2 n + 2] Fibonacci[2 n + 3], {n, 9}]

跟这个扯上关系了——1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465,

\(\frac{B-A}{B}=\frac{5}{10}=\frac{1×5}{2×5},\frac{B-A}{B}=\frac{39}{65}=\frac{3×13}{5×13},\frac{B-A}{B}=\frac{272}{442}=\frac{8×34}{13×34},\frac{B-A}{B}=\frac{1869}{3026}=\frac{21×89}{34×89},\frac{B-A}{B}=\frac{12815}{20737}=\frac{55×233}{89×233},\frac{B-A}{B}=\frac{87840}{142130}=\frac{144×610}{233×610},\frac{B-A}{B}=\frac{602069}{974170}=\frac{377×1597}{610×1597},\frac{B-A}{B}=\frac{4126647}{6677057}=\frac{987×4181}{1597×4181},\frac{28284464}{45765226}=\frac{2584×10946}{4181×10946},\)

{{y -> 5, x -> 10}, {y -> 39, x -> 65}, {y -> 272, x -> 442}, {y -> 1869, x -> 3026}, {y -> 12815, x -> 20737}, {y -> 87840, x -> 142130}, {y -> 602069,  x -> 974170}, {y -> 4126647, x -> 6677057}, {y -> 28284464, x -> 45765226}}
可以有简单的方程——NSolve[{(x - 1) x + x (x + y) == (x + y) (x + y + 1), x > 0, 900000000 > y > 0}, {y, x}, Integers]

\(\displaystyle\prod_{k=A-1}^{B-1}+\prod_{k=A}^{B}≡\prod_{k=A+1}^{B+1}\)——最后出来这个东东！——看似简单！不好找！看官！要不你也来一个？

王守恩

\(x^n+(x+1)^n+(x+2)^n+\cdots+(x+j)^n=y^n+(y+1)^n+(y+2)^n+\cdots+(y+k)^n\)

来个反例！——x, y, j, k, n = 整数。0 ≤ k < j < 799。0 < x < y < 8999。4 ≤ n < 69。

这个代码慢一点。
Do[Catch[Do[If[Sum[(x + i)^n, {i, 0, j}] == Sum[(y + i)^n, {i, 0, k}], Print[n, x, j, y, k]; Throw[True]], {k, 799}, {j, k + 1, 799}, {x, 8999}, {y, x + 1, 8999}]; Print["n=", n, "未找到"]], {n, 4, 69}]
a=4未找到
a=5未找到
a=6未找到

这个代码快一点。
Do[A = Association[]; U = Catch[Do[W = Sum[(x + i)^n, {i, 0, j}]; V = W; If[KeyExistsQ[A, V], B = A[V];
If[B[[1]] != x && B[[2]] != j, Throw[{x, j, B[[1]], B[[2]], W}]], A[V] = {x, j}], {j, 1, 799}, {x, 1, 8999}]; "未找到"];
Print["n=", n, ": ", U], {n, 4, 69}]
n=4: 未找到
n=5: 未找到
n=6: 未找到
n=7: 未找到
n=8: 未找到
n=9: 未找到

我这个软件没用——往前动不了——目标很明确——看软件能否找到反例——再来想想为什么？

王守恩

连续数的积 + 连续数的积 ≡ 连续数的积。——还是无限数字串！——方法有了！答案挺多的！——随便来一个！！

\(4×5×6×7+5×6×7×8=3×4×5×6×7\)

\(5×6×\cdots×14+6×7×\cdots×15=4×5×\cdots×14\)

\(6×7×\cdots×23+7×8×\cdots×24=5×6×\cdots×23\)

\(7×8×\cdots×34+8×9×\cdots×35=6×7×\cdots×34\)

\(8×9×\cdots×47+9×10×\cdots×48=7×8×\cdots×47\)

\(9×10×\cdots×62+10×11×\cdots×63=8×9×\cdots×62\)

{7, 14, 23, 34, 47, 62, 79, 98, 119, 142, 167, 194, 223, 254, 287, 322, 359, 398, 439, 482}——最后一个数。

\(\displaystyle\prod_{k=n+1}^{n^2-2}+\prod_{k=n+2}^{n^2-1}≡\prod_{k=n}^{n^2-2}\)

风花飘飘

王守恩 发表于 2026-8-25 16:34
\(x^n+(x+1)^n+(x+2)^n+\cdots+(x+j)^n=y^n+(y+1)^n+(y+2)^n+\cdots+(y+k)^n\)

来个反例！——x, y, j,  ...

老王，这本质上就是利用了 n^2-1 = (n+1)(n-1) 这个因式分解。
这个哪来的反例？
呵呵