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

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. 
 A: 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.). 
A: 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.
A: 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 $2$.
