Unit fraction representations using shifted Golomb primes

Abstract

We prove that every positive rational number is a sum of distinct unit fractions with denominators of the form $g-h$, where $g$ is a Golomb prime and $h$ is a fixed nonzero integer, as long as the necessary congruence conditions are satisfied. Golomb primes, introduced by Golomb in 1955, are a natural analogue of the primes. Erdős showed that the relative density of Golomb primes among the primes up to $N$ is about $1/\log\log N$, so they form a sparse subset of the primes. Our result is an analogue of Bloom's result on unit fraction representations using shifted primes, for this sparse subset of the primes. A key ingredient in our proof is a Siegel--Walfisz-type theorem for Golomb primes in arithmetic progressions, which we also prove here.