Reasoning about Primes (II)

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.