Underapproximation by Egyptian fractions

Egyptian Fractions Revisited

It is well known that the ancient Egyptians represented each fraction as a sum of unit fractions – i.e., fractions with unit numerators; this is how they, e.g., divided loaves of bread. What is not clear is why they used this representation. In this paper, we propose a new explanation: crudely speaking, that the main idea behind the Egyptian fractions provides an optimal way of dividing the loa...

Binary Egyptian Fractions

Let Ak*(n) be the number of positive integers a coprime to n such that the equation a n=1 m1+ } } } +1 mk admits a solution in positive integers (m1 , ..., mk). We prove that the sum of A2*(n) over n x is both >>x log 3 x and also <<x log x. For the corresponding sum where the a's are counted with multiplicity of the number of solutions we obtain the asymptotic formula. We also show that Ak*(n)...

p-Adic Egyptian Fractions

Denser Egyptian Fractions

An Egyptian fraction is a sum of reciprocals of distinct positive integers, so called because the ancient Egyptians represented rational numbers in that way. In an earlier paper, the author [8] showed that every positive rational number r has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator. More precisely, there exists a p...

Denominators of Egyptian Fractions