Recall that $$\sum_{j=1}^nj^2=\frac{n(n+1)(2n+1)}{6}.$$ When is this quantity a perfect square? It appears that the only solutions are $n=0,1,24.$ By setting $x=12n+6$, the problem reduces to finding rational points on the elliptic curve $$ y^2=x(x+6)(x-6), $$ but I do not know where to go from here. How can I show that these are the only solutions?
2025-01-12 23:56:58.1736726218
When is sum of squares a perfect square?
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