A Symmetry Conjecture for Twin Prime Middles and Its Relation to the Twin Prime Conjecture

Call an integer m a twin middle if m-1 and m+1 are both prime. We observe that for every twin middle m > 348 up to about 4.9 x 10^6 there is d >= 1 such that m-d and m+d are also twin middles; equivalently, the twin sum 2m = (m-1)+(m+1) can be written crosswise from two other twin pairs. The only exceptions found are m = 4, 6, 198, 348 (2m = 8, 12, 396, 696). We state this as a conjecture and show that it implies the twin prime conjecture, so it is at least as hard.

We also give a Hardy-Littlewood-type model R(m) ~ c * S(m) * m/(ln m)^4 for the number R(m) of such representations (correlation 0.955 with computed values for 350 <= m <= 20,000), prove a mod-30 constraint explaining why twin middles with m = +-2 (mod 5) have about three times fewer partners, and show that the exceptions 198 and 348 are of this type and not unlikely under the model. The statement is not claimed to be known to be true or new. Verification code is included.