Golomb's conjecture on prime gaps

Question 10208b (1992) of the American Mathematical Monthly asked: does there exist an increasing sequence $\{a_k\}$ of positive integers and a constant $B > 0$havingthepropertythat${ a_k + n}$containsnomorethan$B$ 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.