Prove $a^k\equiv 1 \pmod k$ has no solution for infinitely many even integers k
123036
Is the equation $x = \phi(n), x=2k, n,k \in \mathbb{Z}$, where $\phi(n)$ is the Euler totient function, solvable for all evens?
135465
Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
46147
Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
32762
Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
24837
$x^2+1=0$ uncountable many solutions
102234
Show $a^p \equiv a \mod 2p$
28148
Are there any simple ways to see that $e^z-z=0$ has infinitely many solutions?
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
63129
Infinite number of $k$ satisfying $p_{k+1} - p_{k} > 2$?
96398
An integer $a \pmod m$ has inverse if and only if $\gcd(a,m)=1$?
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
19071
solution set for congruence $x^2 \equiv 1 \mod m$
79219
Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
81818
Prove or disprove : $\det(A^k + B^k) \geq 0$
155743
Why does the equation $x^2\equiv2 \pmod 5$ have no solutions?