A proof of cases of de Polignac's conjecture

Abstract

For $n \geq 1$ let $ p_n $ denote the $n^{\rm th}$ 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 \geq 0,$ let \\ $T_x := \{ 30x+i \; | \; i \in S\}.$ For each $ x,$ $T_x$ contains at most seven primes. Let $[ \; ]$ denote the floor or greatest integer function. For each integer $s \geq 30$ let $\pi_7(s)$ denote the number of integers $x, \; 0 \leq x < [\frac {s}{30}]$ for which $T_x$ contains seven primes. Let $m \geq 10^{10}$ be an integer and let $P_{K_m}$ denote the largest prime number less than $\sqrt{\prod_{i=1}^{m}p_i}.$ In this paper we show that $$\frac{\prod_{i=1}^{m}p_i}{8(K_m+1)} < \pi_7\left(\prod_{i=1}^{m}p_i\right) $$ and thereby prove that there are infinitely many values of $x$ for which $T_x$ 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 $p'-p = k.$