Several other theorems were proved concerning prime numbers. Many great mathematicians approached problems that are related to primes. There are still many open problems of which we will mention some.

**Conjecture 1**.

__Twin Prime Conjecture__

There are infinitely many pairs primes $p$ and $p + 2$.

**Conjecture 2**.

__Goldbach's Conjecture__

Every even positive integer greater than 2 can be written as the sum of two primes.

**Conjecture 3**.

__The $n^2 + 1$ Conjecture__

There are infinitely many primes of the form $n^2 + 1$, where $n$ is a positive integer.

**Conjecture 4**.

__Polignac Conjecture__

For every even number $2n$ are there infinitely many pairs of consecutive primes which differ by $2n$.

**Conjecture 5**.

__Opperman Conjecture__Is there always a prime between $n^2$ and

$(n + 1)^2$?

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