TITLE: when is $n!+10$ a perfect square?
QUESTION [12 upvotes]: When is $n!+10$ is a perfect square ? I have tried and found that only for $n=3$ is $n!+10$ a perfect square. Is there any other solution to this?

REPLY [22 votes]: A perfect square is congruent to $0$ or $1 \bmod 4$.
If $n\ge 4$ then $n! \equiv 0 \bmod 4$ and so $n!+10 \equiv 2 \bmod 4$ is not a square.
The case $n \le 3$ is dealt by hand.