Denser Egyptian Fractions
Abstract
An Egyptian fraction is a sum of distinct unit fractions (reciprocals of positive integers). We show that every rational number has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator, improving a result from an earlier paper to bestpossible form. We also settle, in bestpossible form, the related problem of how small M_t(r) can be such that there is an Egyptian fraction representation of r with exactly t terms, the denominators of which are all at most M_t(r). We also consider the following problems posed by Erdos and Graham: the set of integers that cannot be the largest denominator of an Egyptian fraction representation of 1 is infinite  what is its order of growth? How about those integers that cannot be the secondlargest (thirdlargest, etc.) denominator of such a representation? In the latter case, we show that only finitely many integers cannot be the secondlargest (thirdlargest, etc.) denominator of such a representation; while in the former case, we show that the set of integers that cannot be the largest denominator of such a representation has density zero, and establish its order of growth. Both results extend to representations of any positive rational number.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 1998
 arXiv:
 arXiv:math/9811112
 Bibcode:
 1998math.....11112M
 Keywords:

 Number Theory;
 11D68
 EPrint:
 26 pages