A proof of cases of de Polignac's conjecture

Abstract

For $n\ge 1$ let $p_n$ denote the $n^{\mathrm{th}}$ 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 $x\ge 0,$ 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 $s\ge 30$ let ${\pi }_7(s)$ denote the number of integers $x,0\le x<[\frac{s}{30}]$ for which $T_x$ contains seven primes. Let $m\ge 10^{10}$ 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 $p^{\prime }$ for which $p^{\prime }-p=k.$

10 pages. arXiv admin note: substantial text overlap with arXiv:1909.02205

References in corpus (1)

• On the Twin Prime Conjecture