解方程 $n^3+(n+1)^3+\cdots+(n+m)^3=p^3$
解方程
\[n^3+(n+1)^3+\cdots+(n+m)^3=p^3\]
若记 $M=n+m$, 且假设 $n,M\in[1,1000]$, 则有以下解:
>> 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.
>> solve_n3plus_until_M3_eq_p3(1000,2000)
in> solve_n3plus_until_M3_eq_p3(1000,2000)
out>
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Total: 0 solutions.
问: 其一般的参数解的形式是怎样的?
Remark: 问题来源于“数学人交流群”.