Towards Proving Legendre's Conjecture

TL;DR

This paper investigates a generalized form of Legendre's conjecture, establishing explicit bounds for the existence of primes in certain intervals using elementary combinatorial methods inspired by Erdős, without relying on the prime number theorem.

Contribution

The paper provides explicit bounds N(k) ensuring primes exist between kn and (k+1)n for all n >= N(k), extending the Bertrand-Chebyshev theorem to a broader context.

Findings

Established explicit bounds N(k) for prime existence in intervals

Extended Bertrand-Chebyshev theorem to multiple intervals

Used elementary combinatorial techniques without prime number theorem

Abstract

Legendre's conjecture states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. We consider the following question : for all integer n>1 and a fixed integer k<=n does there exist a prime number such that kn < p < (k+1)n ? Bertrand-Chebyshev theorem answers this question affirmatively for k=1. A positive answer for k=n would prove Legendre's conjecture. In this paper, we show that one can determine explicitly a number N(k) such that for all n >= N(k), there is at least one prime between kn and (k+1)n. Our proof is based on Erdos's proof of Bertrand-Chebyshev theorem and uses elementary combinatorial techniques without appealing to the prime number theorem.