Let $p$ be a prime number. Prove there exists an integer $a$ such that $p\mid (a^2-a+3)$ iff there exists an integer $b$ such that $p\mid(b^2-b+25)$.
8784
modulo version of the quadratic formula and Euler's criterion
42376
Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$
114867
Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
7491
How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$?
50781
$(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
5981
For which values of $a$ does $a-2 \mid a^3+4$
19979
Why does $p^2+8$ prime imply $p^3+4$ prime
78917
Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$.
64453
Prove if $(a,p)=1$, then $\{a,2a,3a,...,pa\}$ is a complete residue system modulo $p$.
90283
Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$
12365
Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$
20881
Prove that if $p$ and $q$ are distinct primes such that $pq\mid n^2$ then $pq\mid n$.
146512
Prove that for every prime number $p>3$, $4p^2+1$ can be written as the sum of three square numbers
71788
Prove existence of multiplicative inverse modulo p when p is prime
95111
Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
77806
Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
34692
Prove if $ p$ is a prime of the form $3n+1$, then $p$ is also of the form $6m+1$