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Week 6 · ErdősOpen
Primary source

Erdős–Straus Conjecture — Splitting 4/n into Unit Fractions

Simple rule · open problem· Posed 1948

Problem

For every integer n≥2n \ge 2, can 4/n4/n be written as 1/a+1/b+1/c1/a + 1/b + 1/c with positive integers a,b,ca, b, c? E.g., 4/5=1/2+1/4+1/204/5 = 1/2 + 1/4 + 1/20.

Why it matters

Inspired by the ancient Egyptian (Rhind papyrus) tradition of writing fractions as sums of unit fractions. In 1948 Erdős and Straus conjectured: when the numerator is 4, three unit fractions always suffice. Analogous open conjectures exist for numerators 5 (Sierpiński) and 6 (Stewart).

Progress so far

Verified for n≤1017n \le 10^{17} (Salez 2014). Counterexamples are only possible when n≡1,112,132,172,192,232(mod840)n \equiv 1, 11^2, 13^2, 17^2, 19^2, 23^2 \pmod{840} (Schinzel–Mordell) — i.e. only on a density-zero residue class. Explicit formulas are known for almost all nn.

Further reading

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