Conjectures
Erdos-Straus Conjecture
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Number Theory
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A conjecture in number theory formulated in 1948 by Paul Erdos and Ernst Straus, holding that every fraction of the form 4/n can be written as a sum of three positive unit fractions. It remains unsolved, though it has been verified for every n up to 10^17.
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 Cross-Tradition Connections
Sources
1. Erdos-Straus Conjecture (Wikipedia)
WikipediaLead and verification sectionsQuote, Lead and verification sections
The conjecture is named after Paul Erdos and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics.
1. Erdos-Straus Conjecture (Wikipedia)
WikipediaStatement sectionQuote, Statement section
4/n = 1/x + 1/y + 1/z
1. Erdos-Straus Conjecture (Wikipedia)
WikipediaLead sectionQuote, Lead section
an unproven statement in number theory
1. Erdos-Straus Conjecture (Wikipedia)
WikipediaVerification and progress sectionsQuote, Verification and progress sections
Computer searches have verified the truth of the conjecture up to n <= 10^17
1. Erdos-Straus Conjecture (Wikipedia)
WikipediaHistory sectionQuote, History section
who formulated it in 1948
1. Erdos-Straus Conjecture (Wikipedia)
WikipediaIn Branch: Number Theory, Opening paragraphQuote, In Branch: Number Theory, Opening paragraph
The Erdos-Straus conjecture is an unproven statement in number theory.
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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