The Erdős–Straus conjecture

Unsolved Research level Archive

archive posted 6 weeks ago

Prove that for every integer $n \geq 2$ there are positive integers $x, y, z$ with

\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} .

Posed by Paul Erdős and Ernst G. Straus, 1948

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History and references

Still open as of 2026.

Erdős and Straus formulated the conjecture in 1948, and it belongs to the ancient topic of Egyptian fractions, sums of unit fractions like the one above. It is enough to treat prime values of $n$ , because a solution for a divisor of $n$ can be scaled up to a solution for $n$ . Many residue classes of $n$ are covered by explicit formulas, which makes computer searches efficient, and the conjecture has been verified for all $n \leq 1 0^{17}$ . If negative unit fractions are allowed, the statement is known to hold. The general case is open.

References

Erdős–Straus conjecture (Wikipedia)

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