If $p$ is a prime number and $p\mid a^p-b^p$. Then $p^2\mid a^p -b^p$.
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
141088
Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
86594
If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
58133
${a^2}{\mid}({b^3} + 1)\;$and$\;{b^2}{\mid}({a^3} + 1)$
90916
Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
66851
If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
26503
Suppose that $p$ ≥ $q$ ≥ $5$ are both prime numbers. Prove that 24 divides ($p^2 − q^2$)
126893
A group of order 2p (p prime) and other conditions - prove abelian.
84603
Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
94516
If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$
50413
Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$
6332
If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
127293
Prove that $(a-b) \mid (a^n-b^n)$
108237
Show $a^p \equiv b^p \mod p^2$
40968
show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
102234
Show $a^p \equiv a \mod 2p$
34588
Number Theory: $p!+p=n^2$, $p$ = prime