A proof of cases of de Polignac's conjecture

TL;DR

The paper proves infinitely many instances where specific prime patterns occur, including twin primes and cases of de Polignac's conjecture, by analyzing prime distributions within certain arithmetic progressions.

Contribution

It introduces a novel inequality involving prime products and prime counts, establishing infinitely many prime pairs with fixed differences, thus proving key cases of de Polignac's conjecture.

Findings

Proves infinitely many prime pairs with fixed even differences.

Establishes a new inequality linking prime products and prime counts.

Confirms the twin prime conjecture as a special case.

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 $p'-p = k.$