On the Ribenboim-Landau Conjecture

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.