#prime-numbers

Twin primes are pairs of primes that differ by 2, such as $11,13$ or $41,43$. Prove that there are infinitely many twin prime pairs.

Prove that every even integer greater than 2 is the sum of two primes. For example $28=5+23$ and $100=3+97$.

Prove that for every positive integer $n$ there is a prime number strictly between $n^2$ and $(n+1{)}^2$.