Erdős-Straus conjecture

In 1948, Paul Erdős and Ernst Straus conjectured that for an integer n>1 there is always a solution to


where a, b and c are integers in the relation 0<abc. This is the Erdős-Straus . Put another way, 4n can be rewritten as a sum of three unit fractions. The three unit fractions need not be distinct, but some people consider solutions with distinct unit fractions to be more elegant. It is believed that for all n>4 solutions with distinct unit fractions are possible.

As with any conjecture, a single counterexample is enough to disprove, but no multitude of examples is enough to prove. With brute force computer calculations, Allan Swett has obtained examples for all n<1014. Because of the Hasse principle of Diophantine equationsMathworldPlanetmath, we can be sure that for semiprimes pq (where p and q are distinct primes) a solution can be found by looking at 4p or 4q. Researchers are therefore certain that if a counterexample exists, it is surely a prime numberMathworldPlanetmath. Thus Swett has only made available the raw data only for selected prime n rather than for all n he tested.

Title Erdős-Straus conjecture
Canonical name ErdHosStrausConjecture
Date of creation 2013-03-22 16:28:04
Last modified on 2013-03-22 16:28:04
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 8
Author CompositeFan (12809)
Entry type Conjecture
Classification msc 11A67
Synonym Erdös-Straus conjecture
Synonym Erdos-Straus conjectureMathworldPlanetmath
Synonym Erdos-Strauss conjecture