Dense Egyptian Fractions
Abstract
Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest demoninator. We show that for every positive rational there exist Egyptian fractions whose largest denominator is at most N and whose denominators form a positive proportion of the integers up to N, for sufficiently large N; furthermore, the proportion is within a small factor of best possible.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 1998
 DOI:
 10.48550/arXiv.math/9804045
 arXiv:
 arXiv:math/9804045
 Bibcode:
 1998math......4045M
 Keywords:

 Number Theory;
 11D68
 EPrint:
 16 pages