Abstract
We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most solutions of. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time, for any. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation is. Previously, the best known lower bound of this type was of order.
Original language | English |
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Pages (from-to) | 1401 - 1427 |
Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |
Volume | 150 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2020 |
Keywords
- Diophantine equations
- Erdos-Straus equation
- unit fractions
ASJC Scopus subject areas
- Mathematics(all)