Interesting Esoterica

Denser Egyptian Fractions

Article by Martin, Greg
  • Published in 1998
  • Added on
In the collection
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 best-possible form. We also settle, in best-possible 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 Erdős 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 second-largest (third-largest, etc.) denominator of such a representation? In the latter case, we show that only finitely many integers cannot be the second-largest (third-largest, 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.

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BibTeX entry

@article{Martin1998,
	title = {Denser Egyptian Fractions},
	author = {Martin, Greg},
	url = {http://arxiv.org/abs/math/9811112 http://arxiv.org/pdf/math/9811112v1},
	urldate = {2015-07-23},
	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 best-possible form. We also settle, in best-possible 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 Erd{\H{o}}s 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 second-largest (third-largest, etc.) denominator of such a representation? In the latter case, we show that only finitely many integers cannot be the second-largest (third-largest, 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.},
	comment = {},
	collections = {Easily explained},
	year = 1998
}