The Erdős–Straus Conjecture

The Erdős–Straus conjecture is an unsolved problem in number theory. It proposes that every fraction of the form 4/n, where n is an integer greater than or equal to 2, can be expressed as a sum of three positive unit fractions (fractions with a numerator of 1).

For every integer n ≥ 2, there exist positive integers x, y, z such that:
4/n = 1/x + 1/y + 1/z

Historical Context and Origin

The problem belongs to the study of Egyptian fractions, as ancient Egyptians expressed all fractions as sums of distinct unit fractions. The conjecture was formally posed in 1948 by the renowned mathematicians Paul Erdős and Ernst G. Straus.

Current Status of the Conjecture

The conjecture has been verified by computers for all values of n up to at least 10¹⁷. Despite this overwhelming computational evidence, a general mathematical proof valid for all integers n remains elusive. It is considered a major open problem in Diophantine analysis.

A significant simplification is that it is sufficient to prove the conjecture for all prime numbers p. A solution for a prime number guarantees a solution for all of its multiples.

Known Solution Patterns

Mathematicians have discovered explicit polynomial identities that provide solutions for infinite families of numbers. These cover most numbers that are congruent to specific values modulo small primes like 3, 4, 5, 7, or 8. For example:

For any integer n that is congruent to 3 modulo 4, one valid solution is:

4/n = 1/((n+1)/4) + 1/((n+1)/4) + 1/(n(n+1)/4)

The Core Challenge

The principal obstacle to a complete proof involves primes that are quadratic residues modulo small primes. The polynomial methods that solve large classes of numbers fail to cover an infinite set of primes falling into this category. This key obstruction was first noted by mathematician Louis Mordell.

Modern research, including work by Terence Tao and Christian Elsholtz, has shifted toward counting the number of solutions for a given n and studying their asymptotic density, which has been shown to grow logarithmically.

Illustrative Example

For n = 5, the conjecture holds true. Here is one decomposition of 4/5 into three unit fractions:

4/5 = 1/2 + 1/4 + 1/20

An alternative valid decomposition is: 1/2 + 1/5 + 1/10.

Significance

The conjecture's deceptively simple statement connects deeply to fundamental questions in number theory concerning the distribution of prime numbers and the solvability of Diophantine equations. A proof would likely require novel and insightful mathematics.