TITLE: For $n\ge4$, prove that $1!+2!+\cdots+n!$ cannot be the square of a positive integer
QUESTION [5 upvotes]: I'm trying to prove this by induction but seem to be getting nowhere.

REPLY [11 votes]: Every square is either $0,1$ or $4\,\bmod 5$,. For $k\ge 5$ we have $k!\equiv 0\bmod 5$ hence for $n\ge 4$
$$1!+2!+\cdots+n!\equiv 1!+2!+3!+4!\equiv 33\equiv 3\mod 5$$
thus it cannot be a square.