I have written a similar post a while back, but want to highlight this again. We represent odd numbers as $2n+1$. Therefore, all the composites are in the form of $(2n+1)(2m+1)$ = $2(2mn+m+n)+1$. Let $2mn+m+m$ be a.
Therefore, $2a+1$ is a composite when $a=2mn+m+n$ and $2a+1$ is a prime when a is not expressible as $2mn+m+n$. Now $2mn+m+n$ can be written as $(2mn+m)+n$ and $2mn+m=(2n+1)m$.
The only numbers that CANNOT be represented as $(2n+1)m$ is $2^x$. Now, talking about primes as $2a+1$, $a≠2mn+m+n$ and $2^x≠2mn+m$.
That gives us, $2^x+n≠2mn+m+n$ (inequality is maintained here). So, if we have a generalization for n, we can represent primes as $2a+1=2(2^x+n)+1$ i.e. $a=2^x+n$.
Comment:
Due to Fermat little theorem, if $(2, x)=1$ then:
$2^x-2\equiv 0\bmod x$
We rewrite a as:
$a=2^x-2 +n+2$
We want a prime, so we must have:
$[x, (n+2)]=1$
Also: $[(n+2),2]=1$
So general form of n can be:
$n=p+2$
Where p is some particular primes; for example:
$x=3, p=3\Rightarrow a=2^3+3+2=13$
$x=3, p=7\Rightarrow a=2^3+7+2=17$
$x=5, p=3\Rightarrow a=2^5+3+2=37$
$x=5, p=7\Rightarrow a=2^5+7+2=41$
$x=7, p=3\Rightarrow a=2^7+7+2=137$
$x=7, p=19\Rightarrow a=2^7+19+2=149$
There can be other forms too. $n=p+2$ is one form.