Conjectures Involving Sequences and Prime Numbers

Abstract

A 1992 conjecture of Golomb asserts the existence of an infinite increasing sequence $A=\{a_n\}$ of positive integers for which each translate $A_k=\{a_n+k\}$ of $A=A_o$ contains no more than $B$ primes, for some finite bound $B$. This conjecture is inconsistent with the “Prime $k$-tuples Conjecture”, which asserts that for infinitely many positive integers $n$, all $k$ numbers $\{n,n+a_1,n+a_2,...,n+a_{k-1}\}$ are prime, provided that there is no prime number $q$ for which the $k$ integers $\{0,a_1,a_2,...,a_{k-1}\}$ occupy all $q$ residue classes modulo $q$. This paper discusses reasons for believing or disbelieving each of these conjectures.