## A (probably) useless reformulation of the twin prime conjecture

In light of the previous post the following is obviously a long stretch, but I can’t resist mentioning it.

The way Jones-Sato-Wada-Wiens construct their polynomial is to start from the characterization of primes by Wilson’s theorem (namely that $k+1$ is prime iff $k+1$ divides $k!+1$), to convert that into a system of exponential equations, and this is itself converted into a larger system of polynomial equations, which are finally assembled into one big polynomial.

Now twin primes can be characterized also by a Wilson-like statement, namely $k+1, k+3$ are twin primes iff $(k+1)(k+3)$ divides $4(k!+1)+k+1$.

So one could construct a polynomial in several integer variables $P_{2}$ along the same lines, and end up with the (difficult to exploit) reformulation of the twin prime conjecture : there are infinitely many twin primes iff $P_{2}$ takes on arbitrarily high positive values on $\mathbb{N}^m$ (where $m$ is the number of variables). Said differently, a disproof of the conjecture would follow from an absolute bound.

By Polymath8b, a similar polynomial $P_{246}$ for the case of pairs $(n, n+246)$ is known to indeed take arbitrarily high positive values, while of course the same is true for the original Jones-Sato-Wada-Wiens polynomial since there are infinitely many primes.  So those two “sandwich” $P_2$ somehow, and there just might be hope to say educated things about that (which I obviously can’t, but the learned reader is encouraged to do so, should that seem to be a good idea).

Update: for sake of completeness, I should have mentioned that the Wilson-like characterization of twin primes goes back to a paper of Clement in the Montly from 1949, and the analog for pairs of primes $n, n+2k$ has been worked out by Lee and Park to be that $n(n+2k)$ divides $2k(2k)!((n-1)!+1)+((2k)!-1)n$.