PreprintCC BY 4.0arXiv
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/loglog 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.
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BibTeX
@misc{agrawal2026unit,
title = {{Unit fraction representations using shifted Golomb primes}},
author = {Rishika Agrawal},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06798},
eprint = {2611.06798},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06798},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unit fraction representations using shifted Golomb primes
AU - Rishika Agrawal
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06798
UR - https://arxiv.org/abs/2611.06798
AB - 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/loglog 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.
ER -