<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] This is OEIS A031138, which lists some more and says $a(n) =11\cdot(a(n-1)-a(n-2)) + a(n-3) \\ a(n)=-1/2+((3-\sqrt 6)/4)\cdot(5+2\sqrt 6)^n+((3+\sqrt 6)/4)\cdot(5-2\sqrt 6)^n$
</s>