Erdős–Straus Conjecture

The conjecture asks whether $4/n$ is always a sum of three unit fractions: for every integer $n\ge2$ there should be positive integers $x,y,z$ with $4/n=1/x+1/y+1/z$. Erdős raised the question in 1948 in correspondence with Straus, and it entered the literature through the standard collections of unsolved problems in number theory [ErdosStraus1948].

Since a representation for $n$ yields one for every multiple of $n$, it suffices to treat prime $n$. Explicit congruence identities settle all $n$ outside a few thin residue classes, and computer searches have verified the conjecture over enormous ranges. Vaughan showed that the potential exceptions form a set of density zero in a strong quantitative sense [Vaughan1970ErdosStraus], and Elsholtz and Tao studied the finer question of how many representations a typical prime $n$ admits [ElsholtzTao2013].

The structural obstacle is that no finite collection of such identities covers every integer, so the congruence approach cannot by itself close the remaining residue classes. The conjecture remains open; a resolution requires either an argument going beyond covering congruences or a counterexample.