Does this imply infinite twins?

[e2theipi2026]

I can prove the twin prime counting function has this form:

$$\pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1,$$

where $$\pi_2(n)$$ is the twin prime counting function, $$f(n)$$ is the number of twin composites less than or equal to $$n$$ and $$\pi(n)$$ is the prime counting function.

At $$n=p_n,$$ this becomes

$$\pi_2(p_n) = f(p_n) + \pi(p_n) + \pi(p_n + 2) - p_n - 1.$$

With this form, can I make the following argument?: Assume the twin prime counting function becomes a constant $$c$$, then I can change the twin prime counting function to $$c$$ in the equation. The prime counting function $$\pi(n)$$ at the prime sequence $$p_n$$ is just $$n$$, so I can change that to $$n$$. Because I'm assuming no more twin primes, $$p_n+2$$ is not a prime so $$\pi(p_n+2)$$ will also become $$n$$, the equation directly above this paragraph can therefore be simplified to:

$$c = f(p_n) + 2n - p_n - 1.$$

Adding $$1$$ to both sides of this and rearranging it gives,

$$p_n - f(p_n) = 2n - b$$, where $$b=c+1.$$

The right side of $$p_n - f(p_n) = 2n - b$$

has only one possible parity, either odd or even because it is an even number $$2n$$ minus a constant $$b.$$

But, the left side can be both odd and even many times over because $$f(p_n)$$ can be odd or even and is subtracted from $$p_n$$ which is odd for $$p>2.$$

So, the left side will change parity for different values of $$n,$$ while the right side of the equation will remain one parity. Therefore, the two sides cannot be equal for all $$n.$$

This seems to show the twin prime counting function cannot become constant and therefore, there are infinite twin primes. Now assuming I can prove the form of the twin prime counting function given at the beginning of this question, does that argument hold water?

 

[Amer]

What is the function $f(n)$?

 

[e2theipi2026]

What is the function $f(n)$?

$$f(n)$$ was defined at the beginning of the post. If you mean more detail, it is counting the number of "smaller" twin composites $$\le n.$$ So, it is counting all composite $$k\le n$$ such that $$k+2$$ is also composite.
It would seem what I have shown is that the Twin Prime Conjecture is equivalent to proving that $$f(p_n)$$ changes parity an infinite number of times. Any ideas? :)