Conjecture
Erdos-Straus Conjecture
AIR-dosh strows kon-JEK-cher (Paul Erdos, Ernst G. Straus)
Number Theory
The Erdos-Straus conjecture is an unproven number theory statement asserting that for every integer n greater than or equal to 2, the fraction 4/n can be expressed as a sum of three positive unit fractions. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor every integer n greater than or equal to 2, there exist positive integers x, y and z such that 4/n = 1/x + 1/y + 1/z. 1 Proposed Year Prize StatusNot a Millennium Prize Problem; no major institutional prize is attached, though it has attracted sustained attention from number theorists since 1948. 1 Progress Toward ResolutionComputer searches have verified the conjecture for every n up to 10^17. Modular identities supply solutions for infinitely many arithmetic progressions of n, and the natural density of potential counterexamples has been shown to be zero, though no general proof exists. Allowing negative integers for x, y or z makes the problem trivially solvable, so the difficulty is specific to the positive-integer requirement. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Additional Source Erdos-Straus Conjecture (Wikipedia)Lead section
Open Questions
Source Erdos-Straus Conjecture (Wikipedia)
Posed By
Source Erdos-Straus Conjecture (Wikipedia)
Source Erdos-Straus Conjecture (Wikipedia)
Sources
1. Erdos-Straus Conjecture (Wikipedia)
WikipediaLead section
Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics.
Background and history
Richard Obláth also published an early work on the conjecture, a paper written in 1948 and published in 1950, in which he extended earlier calculations of Straus and Harold N.
Lead section
an unproven statement in number theory
History section
who formulated it in 1948
View the Source Open Questions (1 open question)
Does the Erdos-Straus equation 4/n = 1/x + 1/y + 1/z really have a positive-integer solution for every integer n of 2 or more?
Computer search has confirmed the equation solvable for every n up to 10^17, and modular identities cover infinitely many residue classes, but no argument closes the remaining, conjecturally empty, set of exceptions for every n at once.
What would resolve this A general proof, or a genuine counterexample, covering every integer n rather than a further extension of the verified range.
Number theoryErdos-Straus Conjecture (Wikipedia)
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