Reasoning about Primes (II)

TL;DR

This paper proves Legendre's and Andrica's conjectures using a novel algorithm that determines the maximum prime gap in a specific combinatorial game, leading to new insights about prime distribution.

Contribution

It introduces an algorithm to exactly bound prime gaps in a combinatorial setting, facilitating proofs of longstanding prime conjectures.

Findings

Proves Legendre's conjecture within the specified interval.

Establishes bounds on prime gaps in the prime game.

Shows the existence of a prime within n ± (sqrt(n)-1) for all n > 1.

Abstract

We prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes <= p in their doubled primorial interval 0..p#..2p# where we relax a constraint that the primes usually follow: if the bound g(P)=2p-5 for maximizing the gap length applies with more degrees of freedom, it also applies in the more constrained prime game as g(P)<=2p-5, at least in the subregion $[1,{p'}^2]$ where no other primes have influence (p' notates the next other prime). From here proving the mentioned theorems is straightforward, for example Legendre's interval $]n^2,(n+1)^2[$ is located completely inside the valid subregion and is greater than the greatest possible gap. Another consequence is that there must be a prime within n+-(sqrt(n)-1) for all n>1. For small numbers the proofs are verified using the R statistical language.