Dense Egyptian fractions

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. (I have since improved these results in the sequel paper Denser Egyptian fractions.)

A description of this work appeared in the October 10, 1997 volume of Science, in an article by Dana Mackenzie called "Fractions to make an Egyptian scribe blanch" (page 224).

Two examples: using denominators less than 1000, I found a representation of 2 as the sum of the reciprocals of 366 distinct integers, and a representation of 6 as the sum of the reciprocals of 453 distinct integers.

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