A lower bound for the number of Egyptian fractions
Year: 2026
Type: article
Abstract: An Egyptian fraction is a sum of the form 1 / n 1 + ⋯ + 1 / n r 1/n_{1} + \cdots + 1/n_{r} where n 1 n_{1} , …, n r n_{r} are distinct positive integers. We prove explicit lower bounds for the cardinality of the set E N {E}_{N} of rational numbers that can be represented by Egyptian fractions with denominators not exceeding N N . More precisely, we show that for every integer k ≥ 4 k \geq 4 such that ln k N ≥ 3 / 2 \ln _{k} N \geq 3/2 it holds ln | E N | ln 2 ≥ ( 2
Source: Mathematics of Computation
DOI: 10.1090/mcom/4190
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", Politecnico di Torino, European Commission, Ministero dell’Istruzione, dell’Università e della Ricerca, Università degli Studi di Genova, Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni, NextGenerationEU
Awards: G4757467837, G490430286
OpenAlex ID: W7123352468