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Sums of four and more unit fractions and approximate parametrizations
Author(s) -
Elsholtz Christian,
Planitzer Stefan
Publication year - 2021
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12452
Subject(s) - mathematics , divisor (algebraic geometry) , set (abstract data type) , unit (ring theory) , upper and lower bounds , rational number , key (lock) , point (geometry) , combinatorics , discrete mathematics , mathematical analysis , geometry , ecology , computer science , biology , programming language , mathematics education
We prove new upper bounds on the number of representations of rational numbers m n as a sum of four unit fractions, giving five different regions, depending on the size of m in terms of n . In particular, we improve the most relevant cases, when m is small, and when m is close to n . The improvements stem from not only studying complete parametrizations of the set of solutions, but simplifying this set appropriately. Certain subsets of all parameters define the set of all solutions, up to applications of divisor functions, which has little impact on the upper bound of the number of solutions. These ‘approximate parametrizations’ were the key point to enable computer programmes to filter through a large number of equations and inequalities. Furthermore, this result leads to new upper bounds for the number of representations of rational numbers as sums of more than four unit fractions.