Primes, Pi, and Irrationality Measure

TL;DR

This paper explores the relationship between primes, the number pi, and irrationality measures, providing a quantified version of Euclid's theorem and deriving bounds on prime gaps using irrationality measures.

Contribution

It introduces a novel approach to bounding prime gaps by applying irrationality measures to a folklore proof of Euclid's theorem.

Findings

Derived a weaker upper bound on prime gaps using irrationality measures.

Connected irrationality measures of specific constants to prime distribution.

Provided a new perspective linking irrationality measures and prime number theorems.

Abstract

A folklore proof of Euclid's theorem on the infinitude of primes uses the Euler product and the irrationality of $\zeta(2) = \pi^2/6$. A quantified form of Euclid's Theorem is Bertrand's postulate $p_{n+1} < 2p_n$. By quantifying the folklore proof using an irrationality measure for $6/\pi^2$, we give a proof (communicated to Paulo Ribenboim in 2005) of a much weaker upper bound on $p_{n+1}$.