Prove: The integer p-1 is a quadratic residue of an odd prime p if and only if p congruent 1 ( mod4).  That’s right?!
I came across this problem and I believe Lagrange's theorem is the key to its solution.  The question is:     Let $p$ be an odd prime. Prove that there is some integer $x$ such that $x^2 \equiv −1 \pmod p$ if and only if $p \equiv 1 \pmod 4$.   I appreciate any help.  Thanks.
A student of mine has been self-studying some elementary number theory. She came by my office today and asked if I had any hints on how to prove the statement      If $m$ is odd then $\gcd(2^m-1,2^n+1)=1$.    It's been a while since I took number theory and I'm not sure what to do. She said she is learning about congruences, primitive roots, and power residues.  She has not taken any group theory.