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

TL;DR

This paper investigates ancient Egyptian fraction problems, proving conditions under which the fraction 4/n can or cannot be expressed as a sum of three distinct unit fractions, advancing understanding of the Erdos-Strauss conjecture.

Contribution

The paper introduces four theorems, including criteria based on prime divisors for the existence of such decompositions, providing explicit formulas when they exist.

Findings

If 4/n cannot be expressed as three distinct unit fractions, then all prime divisors of n are congruent to 1 modulo 24.

If n has a divisor not congruent to 1 modulo 24, then a decomposition exists and is explicitly constructed.

The work advances partial results related to the Erdos-Strauss conjecture and ancient Egyptian fraction representations.

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.