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.

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