Some Conjecture of Prima Number

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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