Can anyone come up with an interesting consequence of the Twin Prime Conjecture being true?

Question

The question is in the title. Was wondering if there are statements equivalent to or a consequence of the statement that there are infinitely many twin primes.

If not, then why is this conjecture a "terminal point" in mathematics considered interesting?

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If this is not an easy question to answer, I am willing to accept known equivalent statements (or consequences). The most elegant one wins.

I have a preference for algebraic statements over analytical. The analytical statements are the majority of published attempts. I dream of their being an algebraic approach.

Answer 1

The twin prime problem is the tip of an iceberg. Settling it might help us decide whether, for all even $k$, there are infinitely many pairs of primes differing by $k$, even whether there are infinitely many pairs of consecutive primes differing by $k$, and that might shed light on the question of whether for every admissible $m$-tuple $(a_1,a_2,\dots,a_m)$ there are infinitely many $n$ such that all of the numbers $n+a_1,n+a_2,\dots,n+a_m$ are prime, and that might give us some insight into Schinzel's Hypothesis H (q.v.).

Answer 2

Don't know what application it will serve in the real world but long back out of curiosity, I wanted to find the asymptotic expansion of the $n$-th twin prime $q_n$ assuming the twin prime conjecture. I got something like

$$ q_n \sim \frac{n\log^2 n}{C}\bigg(1 + \frac{2\log\log n - 1}{\log n - 2}\bigg)^2 $$ where $C$ is twice the twin prime constant.

Answer 3

Zhang's remarkable paper on infinitely many pairs of primes differing by less than $7 \times 10^6$ was brought all the way down to $246$ in a ploymath project initiated by Terrence Tao and sharpened by the latest works of James Maynard. The twin prime conjecture will imply that the bound can be further brought down form $246$ to $2$ which in turn would imply that there is a wealth of mathematical tools waiting to be discovered to be able to reduce the bound form $246$ to $2$.