EGYPTIAN FRACTIONS WITH ODD DENOMINATORS

Denominators of Egyptian Fractions

A Note on Farey Fractions with Odd Denominators

Abstract. In this paper we examine the subset FQ,odd of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of the values taken on by ∆ = qa − aq where a/q < a/q are consecutive in FQ,odd. After proving an asymptotic result for these frequencies, we generalize the result to the subset of elements of FQ,odd formed by res...

On Unit Fractions with Denominators in Short Intervals

In this paper we pove that for any given rational r > 0 and all N > 1, there exist integers N < x 1 < x 2 < · · · < x k < e r+o(1) N such that r = 1 x 1 + 1 x 2 + · · · + 1 x k .

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)...