<s>[INST] When is $1^5 + 2^5 + \ldots + n^5$ a square? When is $1^5 + 2^5 + \ldots + n^5$ a square? I found that this happens sometimes: $n=13$ gives $1001^2$, $n=133$ gives $9712992^2$ and $n=1321$ gives $942162299^2$.
I feel that the identity$$\displaystyle\sum_{i=1}^n i^5 = \tfrac{1}{12}[2n^6+6n^5+5n^4-n^2]$$ will be useful, since it's all square powers except one.. but I see no way to connect that.
 [/INST] If $$a=1+2+3+\cdots+n$$ then
$$
1^5+2^5+3^5+\cdots+n^5 = \frac{4a^3-a^2}{3},
$$
so now the question is: when is that a square?
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