Golomb's conjecture on prime gaps

TL;DR

This paper disproves Golomb's conjecture by showing that no such sequence with bounded primes in all shifts exists, using recent advances in prime number theory.

Contribution

It provides a negative resolution to Golomb's conjecture, leveraging recent progress in prime number theory.

Findings

No increasing sequence of positive integers with bounded primes in all shifts exists.

The conjecture is false due to recent prime number theory results.

The result settles a long-standing question from the American Mathematical Monthly.

Abstract

Question 10208b (1992) of the American Mathematical Monthly asked: does there exist an increasing sequence $\{a_k\}$ of positive integers and a constant $B > 0$ having the property that $\{ a_k + n\}$ contains no more than $B$ primes for every integer $n$? A positive answer to this question became known as Golomb's conjecture. In this note we give a negative answer, making use of recent progress in prime number theory.