2014/11/20 by Jens Oehlschlägel, Oehlschlägel, Jens
Computer Science · Mathematics · #11B05 #Artificial Intelligence in Games #Computability, Logic, AI Algorithms #FOS: Mathematics #G.2 #General Mathematics (math.GM) #Probability and Statistical Research
paper · pdf · doi:10.48550/arxiv.1411.6582
openalex publication_date 2014/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ]n2,(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.