Upper bounds for prime gaps related to Firoozbakht's conjecture

Abstract

We study two kinds of conjectural bounds for the prime gap after the k-th prime $p_k$: (A) $p_{k+1}<(p_k{)}^{1+1/k}$ and (B) $p_{k+1}-p_k<{\log }^2{p}_k-\log p_k-b$ for $k>9$. The upper bound (A) is equivalent to Firoozbakht's conjecture. We prove that (A) implies (B) with $b=1$; on the other hand, (B) with $b=1.17$ implies (A). We also give other sufficient conditions for (A) that have the form (B) with $b\to 1$ as $k\to \infty$.

8 pages, with Corrigendum

References in corpus (3)

• Verification of the Firoozbakht conjecture for primes up to four quintillion
• The distribution of maximal prime gaps in Cramer's probabilistic model of primes
• On a new property of primes that leads to a generalization of Cramer's conjecture