A proof of cases of de Polignac’s conjecture

For n ≥ 1 let pn denote the nth prime number. Let S = {1, 7, 11, 13, 17, 19, 23, 29}, the set of positive integers which are both less than and relatively prime to 30. For x ≥ 0, let Tx := {30x + i | i∈S}. For each x, Tx contains at most seven primes. Let [ ] denote the floor or greatest integer function. For each integer s ≥ 30 let π7 (s) denote the number of integers x, 0 ≤ x < [s/30] for which Tx contains seven primes. Let m ≥ 1010 be an integer and let PKm denote the largest prime number less than √Pmi=1 pi. In this paper we show that Pmi=1 pi/8(Km+1) < p7 (Pmi=1 pi) and thereby prove that there are infinitely many values of x for which Tx contains seven primes. This, in particular, proves the well known twin prime conjecture as well as several cases of Alphonse de Polignac’s conjecture that for every even number k, there are infinitely many pairs of prime numbers p and p’ for which p’ – p = k.