On the Ribenboim-Landau Conjecture

TL;DR

This paper proves the Ribenboim hypothesis, linking the distribution of consecutive primes with the solvability of Landau's problem, thus advancing understanding of prime gaps and their implications.

Contribution

It provides a proof of the Ribenboim hypothesis, establishing a connection between prime gaps and Landau's problem for the first time.

Findings

Proves the Ribenboim hypothesis.

Links prime gap conditions to Landau's problem.

Offers new insights into prime distribution and number theory.

Abstract

We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a positive solution.