TY - JOUR
T1 - The number of solutions of the Erdos-Straus Equation and sums of k unit fractions
AU - Elsholtz, Christian
AU - Planitzer, Stefan
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
KW - Diophantine equations
KW - Erdos-Straus equation
KW - unit fractions
UR - http://www.scopus.com/inward/record.url?scp=85060882292&partnerID=8YFLogxK
U2 - 10.1017/prm.2018.137
DO - 10.1017/prm.2018.137
M3 - Article
AN - SCOPUS:85060882292
SN - 0308-2105
VL - 150
SP - 1401
EP - 1427
JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
IS - 3
ER -