Let a be an integer greater than or equal to 2. If a | mn, then a | m or a | n. Show that a is a prime.
101637
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
107502
Let $p \geq 2$. Prove that if $2^p-1$ is prime then $p$ must also be prime.
119282
If $a_1,a_2,\ldots, a_n$ are distinct primes, and $a_1=2$, and $n>1$, then $a_1a_2\cdots a_n+1$ is of the form $4k+3$.
27544
With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
131385
How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
78917
Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$.
157766
Are primes (ignoring $2$) equally likely to be $1~\text{or}~3\pmod 4$?
60804
About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
140294
prove that $n >2$ and $n$ is a prime then $n$ is odd
24988
Prove there exists $m > 2010$ such that $f(m)$ is not prime
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
32618
Suppose $p$ is a prime number such that $(p-1)/4$ and $(p+1)/2$ are also primes. Show that $p=13$
58935
If $p$ and $p^2+8$ are both prime numbers, then $p^3+4$ is prime.
158563
If $m^4+4^n$ is prime, then $m=n=1$ or $m$ is odd and $n$ even
34954
If $11..11$($n$ times 1) is prime, prove that $n$ is also prime
123033
Showing a number $n$ is prime if $(n-2)! \equiv 1 \pmod n$