TITLE: On primes between $k$ and $k!$
QUESTION [6 upvotes]: I have the following homework question:
"Show that for $k\geq 4$ between $k$ and $k!$ there always exists a prime number of the form $4n+3$."
How can one prove it?

REPLY [5 votes]: Hint: Look at the prime divisors of $k!-1$, and note that $k!-1\equiv -1(\bmod. 4)$ for $k\geq 4$.