TITLE: $1!+2!+\ldots+n!$ cannot be the square of a positive integer
QUESTION [10 upvotes]: I have to prove that $1!+2!+\ldots+n!$ cannot be the square of a positive
integer, $\forall n\geq4$.
I've tried to do this with induction, but I don't seem to reach any satisfactory conclusion.
Any hint will do.

REPLY [4 votes]: See that 1 ! + 2 ! + 3 ! + 4 !=33. And from 5! onwards all the other numbers will end with 0 as its last digit. 
So when you take the sum upto n! with n greater than equal to 4 you can see that the last digit of the sum will be 3. 
So there is no postive integer whose square ends up with 3 as the last digit or unit place's digit. So the sum cannot be a square of a positive integer.