An ancient Egyptian problem:the diophantine equation 4/n=1/x+1/y+1/z, n>or=2

Abstract

From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that has come down to us in the last 60 years, is known as the Erdos- Strauss conjecture which states that for each positive integer n>1; the fraction 4/n can be decomposed into a sum of three distinct unit fractions. Since 1950, a numberof partial results have been achieved, see references [1]- [8]; and also [10] and[11]. In this work we contribute four theorems. In Theorem 2, we prove that if the fraction 4/n is not equal to a sum of three distinct unit fractions, then each prime divisor of n; must be congruent to 1 modulo24. Moreover, if n contains a divisor noncongruent to 1mod24; then such a decomposition does exist and it is explicitly given.