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From the Rhind Papyrus and other extant sources, we know that the ancientEgyptians were very iterested in expressing a given fraction into a sum of unitfractions, that is fractions whose numerators are equal to 1. One of theproblems 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; thefraction 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. InTheorem 2, we prove that if the fraction 4/n is not equal to a sum of threedistinct unit fractions, then each prime divisor of n; must be congruent to 1modulo24. Moreover, if n contains a divisor noncongruent to 1mod24; then such adecomposition does exist and it is explicitly given.

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