问题

\begin{aligned}
3^3 + ... + 5^3 &= 6^3,\\
3^3 + ... + 22^3 &= 40^3,\\
6^3 + ... + 30^3 &= 60^3,\\
6^3 + ... + 69^3 &= 180^3,\\
11^3 + ... + 14^3 &= 20^3,\\
15^3 + ... + 34^3 &= 70^3,\\
\end{aligned}

\(\sum_{k=1}^{n^3}(\frac{n^4-3n^3-2n-2}{6}+k)^3=(\frac{n^5+n^3-2n}{6})^3\)

利用 Calculator 计算.

>> solve_n3plus_until_M3_eq_p3(1,1000)
in> solve_n3plus_until_M3_eq_p3(1,1000)
out>
3^3 + ... + 5^3 = 6^3
3^3 + ... + 22^3 = 40^3
6^3 + ... + 30^3 = 60^3
6^3 + ... + 69^3 = 180^3
11^3 + ... + 14^3 = 20^3
11^3 + ... + 109^3 = 330^3
15^3 + ... + 34^3 = 70^3
34^3 + ... + 158^3 = 540^3
213^3 + ... + 365^3 = 1581^3
213^3 + ... + 555^3 = 2856^3
273^3 + ... + 560^3 = 2856^3
291^3 + ... + 339^3 = 1155^3
406^3 + ... + 917^3 = 5544^3
556^3 + ... + 654^3 = 2805^3
646^3 + ... + 798^3 = 3876^3

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Total: 15 solutions.

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>> solve_n3plus_until_M3_eq_p3(10000,11000)
in> solve_n3plus_until_M3_eq_p3(10000,11000)
out>

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Total: 0 solutions.

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References:

等幂和问题(5)—— 3³+4³+5³=6³ 绝非巧合！ - 知乎 (zhihu.com)