Unit fractions with shifted prime denominators

Thomas Bloom

doi:10.1017/prm.2024.42

Unit fractions with shifted prime denominators

Thomas Bloom

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that any positive rational number is the sum of distinct unit fractions with denominators in {p−1:p prime}. The same conclusion holds for the set {p−h:p prime} for any h∈Z∖{0}, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

Original language	English
Pages (from-to)	1-11
Number of pages	11
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
DOIs	https://doi.org/10.1017/prm.2024.42
Publication status	Published - 2 Apr 2024

Keywords

number theory
prime numbers
unit fractions

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10.1017/prm.2024.42Licence: CC BY

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title = "Unit fractions with shifted prime denominators",

abstract = "We prove that any positive rational number is the sum of distinct unit fractions with denominators in \{p−1:p prime\}. The same conclusion holds for the set \{p−h:p prime\} for any h∈Z∖\{0\}, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.",

keywords = "number theory, prime numbers, unit fractions",

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doi = "10.1017/prm.2024.42",

language = "English",

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N2 - We prove that any positive rational number is the sum of distinct unit fractions with denominators in {p−1:p prime}. The same conclusion holds for the set {p−h:p prime} for any h∈Z∖{0}, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

AB - We prove that any positive rational number is the sum of distinct unit fractions with denominators in {p−1:p prime}. The same conclusion holds for the set {p−h:p prime} for any h∈Z∖{0}, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

KW - number theory

KW - prime numbers

KW - unit fractions

UR - https://www.scopus.com/pages/publications/85189536786

U2 - 10.1017/prm.2024.42

DO - 10.1017/prm.2024.42

M3 - Article

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SP - 1

EP - 11

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

ER -

Unit fractions with shifted prime denominators

Abstract

Keywords

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