Let $p$ be an odd prime and let $m = 2p$. Prove that $a^{m−1} ≡ a \pmod{m}$ for all natural numbers $a$.
102234
Show $a^p \equiv a \mod 2p$
10105
Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule]
152039
Prove that $a^{2^n}=1 \mod 2^{n+2}$
12862
Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd
119630
For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
140198
Let $a$ be a quadratic residue modulo $p$. Prove that the number $b\equiv a^\frac{p+1}{4} \mod p$ has the property that $b^2\equiv a \mod p$.
116686
$p=2^n+1$. Prove that every quadratic nonresidue modulo $p$ is a primitive root modulo $p$
44729
Prove $g$ is a generator if $g^q=1 \pmod p$
7085
Prove that if the equation $x^{2} \equiv a\pmod{pq}$ has any solutions, then it has four solutions.
106401
Prove: $p-mq | f(m)$ where $m$ is any integer
83752
If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$
141369
If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$. Demonstration.
130300
Proof that every quadratic residue has two roots, modulo a prime
45037
Show $(2^m-1,2^n+1)=1$ if $m$ is odd
36941
If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$?
145041
Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
54286
Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$