TITLE: Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$
QUESTION [16 upvotes]: Possible Duplicate:
How to prove that the number 1!+2!+3!+…+n! is never square? 

Show that $\displaystyle\sum\limits_{i=1}^k  i!$ is never a perfect square for $k\ge4$
I could prove $k!$ is never a perfect square using Bertrand's Postulate. But this one seems to be an uphill task.

REPLY [3 votes]: Hint $\rm\ mod\ 5\!:\ \ \mathbb Z^2 \equiv \{0,\: \pm 1,\:\pm2\}^2 \equiv \{0,\: \pm 1\} \not\ni -2 \equiv  1!+2!+3!+4! + 5 N$