For any $a \in \Bbb{Z}$, can we always find two prime numbers $p, q$, such that $p - q \in (a)$?
92737
primes whose difference is a multiple of $n$
59261
If $n\in\Bbb Z^+$ is not a square, prove exist infinitely many primes $p$ such that $\left(\frac{n}{p}\right)=-1$.
50663
prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
75733
Divisibility of 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 $a$
834
If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.
149661
For $n=pq$, show that there exist $\alpha, \beta \in Z_n^*$ such that $\alpha, \beta \notin (Z_n^*)^2$ and $\alpha \cdot \beta \notin (Z_n^*)^2$.
91935
If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$
252
Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?
83643
Are there prime gaps of every size?
70463
Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
156940
There does not exist a polynomial $p(x)$ with integer coefficients which gives a prime number $\forall x\in \mathbb{Z}$
17233
are two consecutive numbers relatively prime?
100426
Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
151701
Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?
78917
Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$.