Large gaps between Romanoff numbers
Abstract
Questions concerning representations of integers as the sum of a prime and a power of two go back to the correspondence between Euler and Goldbach in 1752 and to de Polignac's work of 1849. Romanoff proved that the set of integers admitting such a representation has positive lower density. Following the complementary direction studied by Kalmynin and Konyagin, we consider long intervals containing no Romanoff numbers. We use truncated products of divisor sums to derive the bound Gℛ(X)≫loglog X for the longest block in [1,X] containing no integer of the form p+2n, with p prime and n≥1. This improves the bound Gℛ(X)≫loglog X/logloglog X of Kalmynin and Konyagin. The key step extends a translate lemma to sets of O(log X) integer shifts of absolute value at most X: auxiliary primes dividing differences of shifts are removed at negligible cost. After preliminary sieving, this allows all remaining shifts to be covered simultaneously. The argument is unconditional and also applies to p+an for every fixed integer a≥2.
Related
BibTeX
@misc{radomskii2026large,
title = {{Large gaps between Romanoff numbers}},
author = {Artyom Radomskii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38408},
eprint = {2609.38408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38408},
}
RIS
TY - UNPB
TI - Large gaps between Romanoff numbers
AU - Artyom Radomskii
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38408
UR - https://arxiv.org/abs/2609.38408
AB - Questions concerning representations of integers as the sum of a prime and a power of two go back to the correspondence between Euler and Goldbach in 1752 and to de Polignac's work of 1849. Romanoff proved that the set of integers admitting such a representation has positive lower density. Following the complementary direction studied by Kalmynin and Konyagin, we consider long intervals containing no Romanoff numbers. We use truncated products of divisor sums to derive the bound G_ℛ(X)≫loglog X for the longest block in [1,X] containing no integer of the form p+2^n, with p prime and n≥1. This improves the bound G_ℛ(X)≫loglog X/logloglog X of Kalmynin and Konyagin. The key step extends a translate lemma to sets of O(log X) integer shifts of absolute value at most X: auxiliary primes dividing differences of shifts are removed at negligible cost. After preliminary sieving, this allows all remaining shifts to be covered simultaneously. The argument is unconditional and also applies to p+a^n for every fixed integer a≥2.
ER -