Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
32762
Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
131488
Show that $p^3+4$ is prime
70066
Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
119630
For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
145041
Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
101057
Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
6332
If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
49217
How do I show that $a^p + b^p > (a + b)^p$ for $a,b>0$ and $0<p<1$?
134348
When is $(p - 1)! + 1$ a power of $p$?
74512
How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
121890
Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
12365
Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$
157884
For any prime $p>3$ show that 3 divides $2p^2+1$
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 $b^2\equiv a \mod p$.
94017
If p is a prime and a ∈Z. How do we prove that the congruence x^x ≡ a (mod p) is solvable?
88656
For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$