Erdős–Straus conjecture

In number theory, the Erdős–Straus conjecture states that for all integers $n \geq 2$ , the rational number $4/ n$ can be expressed as the sum of three unit fractions. Paul Erdős and Ernst G. Straus formulated the conjecture in 1948. It is one of many conjectures by Erdős.

More formally, the conjecture states that, for every integer $n \geq 2$ , there exist positive integers $x$ , $y$ , and $z$ such that

For instance, for $n = 5$ , there are two solutions:

Some researchers additionally require these integers to be distinct from each other, while others allow them to be equal. For $n \geq 3$ , it does not matter whether they are required to be distinct: if there exists a solution with any three integers $x$ , $y$ , and $z$ then there exists a solution with distinct integers. For $n = 2$ , however, the only solution is $4/2 = 1/2 + 1/2 + 1/1$ , up to permutation of the summands. When $x$ , $y$ , and $z$ are distinct then these unit fractions form an Egyptian fraction representation of the number $4/ n$ .

The restriction that $x$ , $y$ , and $z$ be positive is essential to the difficulty of the problem, for if negative values were allowed the problem could always be solved. Also, if $n$ is a composite number, $n = pq$ , then an expansion for $4/ n$ could be found from an expansion for $4/ p$ or $4/ q$ . Therefore, if a counterexample to the Erdős–Straus conjecture exists, the smallest $n$ forming a counterexample would have to be a prime number, and it can be further restricted to one of six infinite arithmetic progressions modulo $840$ . Computer searches have verified the truth of the conjecture up to $n \leq 10 17$ , but proving it for all $n$ remains an open problem.

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