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I noticed that $\sum_{i=0}^{24} i^2 = 70^2$. Are there any other solutions to $\sum_{i=0}^{n} i^2 = x^2$, besides the trivial cases where n and x are both 0 or 1? Or, is there a proof that no other solutions exist?

I searched the OEIS and found nothing, so I assume that no further solutions are known.

By looking at the sum in a modulo, m, I can eliminate certain values of n. I use:

$2n^3+3n^2+n$ mod $m$ = $6x^2$ mod $m$

For example, $6x^2$ mod $17$ produces 0, 3, 5, 6, 7, 10, 11, 12, and 14. For n to be a candidate value to solve the equation, $2n^3+3n^2+n$ mod $17$ must produce one of those values. For n = 2 mod 17, the result is $2*2^3+3*2^2+2$ mod $17 = 13$, so no n = 2 mod 17 can result in a perfect square.

I use this to build a "filter" by listing every value in mod m that can be a perfect square. By overlapping many filters, each using a different modulus, I can check numbers very quickly. Every filter I add reduces the number of candidates I have to check, but no finite number of filters will prove that no other solutions exist, since the filters will all align at n = LCM(all filter moduli) and 0, 1, and 24 are all valid candidates in every filter.

So far, I've tried everything up to $n = 10^{12}$ and have not found any other solutions. Either there are no more solutions or the next solution is very large.

Bryan Wolf
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    there are no more solutions. Proof is usually by elliptic curves. see https://math.stackexchange.com/questions/74265/the-sum-of-the-first-n-squares-is-a-square-a-system-of-two-pell-type-equation – Will Jagy Oct 11 '22 at 18:31
  • Not always mentioned, this is a crucial fact in the Leech lattice. See, for example, page 130 in Lattices and Codes by W. Ebeling. – Will Jagy Oct 11 '22 at 20:03

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