How do I prove that $p = x^2 +2y^2 $ iff p $\equiv$ 1 $\mod{8}$ or p $\equiv$ 3 $\mod{8}$?
124995
$p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$
77816
Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
98871
How to prove that if a prime divides a Fermat Number then $p=k\cdot 2^{n+2}+1$?
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
85287
Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$
146053
Proving identities (mod $pq$) using Fermat's little theorem?
21629
Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
45454
Pairs of $(p,q)$ such that $p^2 + 1 = 2 q^2$?
94017
If p is a prime and a ∈Z. How do we prove that the congruence x^x ≡ a (mod p) is solvable?
19979
Why does $p^2+8$ prime imply $p^3+4$ prime
4129
Solvability of $a \equiv x^2 \mod b$
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
79219
Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
45171
How prove this $(p-1)!\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\right)\equiv 0\pmod{p^2}$
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
6160
How to prove a number $a$ is quadratic residue modulo $n$?
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.