Erdős–Straus Conjecture

Can 4/n always be written as a sum of exactly three unit fractions?

Posed by Paul Erdős and Ernst Straus, 1948.

In plain terms

The ancient Egyptians wrote fractions as sums of distinct unit fractions — 1/2, 1/3, 1/7 and so on. The conjecture says that for every whole number n ≥ 2, the fraction 4/n can be written as a sum of exactly three of them.

For n = 5: 4/5 = 1/2 + 1/4 + 1/20. For most n it is easy, and there are simple formulas that handle whole families of n at once.

What is known

It has been verified by computer far past 10¹⁷ — nobody seriously doubts it is true. It is enough to prove it for prime n, since composite cases follow.

The classical approach is identities: algebraic formulas that solve 4/n for every n in some residue class. A dozen such families together handle 834 of the 840 classes modulo 840. The six that resist are exactly the perfect squares among the units mod 840 — a fact traced to Mordell, and not a coincidence.

Why it is hard

Mordell's obstruction is the crux, and it is a genuine barrier rather than a gap in effort. Any solution built from finitely many polynomial identities provably cannot cover those six classes. The identity method does not merely fail there; it cannot succeed. So a proof needs a new mechanism for those residues, and the classes that resist are precisely the ones defined by quadratic-residue conditions — the same structure that makes so many questions about primes hard.

What a breakthrough would mean

Very little outside number theory, and it is worth being blunt about that. There is no cryptographic angle here and no application waiting.

The value is in the mechanism. This is a problem where we know exactly which tool fails and why, so a proof must introduce something genuinely new for handling quadratic-residue obstructions in covering arguments. That machinery would likely transfer to the wider family of Erdős–Graham questions about Egyptian fractions.

There is also a quieter question the search data speaks to directly: not whether 4/n has a representation, but how many it has, and why some primes have so few. That statistical question is tractable now, and it is where this lab's work on the problem actually landed.

What this lab tried

Standing: Evidence Low-f primes are identity-poor: the size-normalized bottom 2% is 98% inside Mordell's six QR classes mod 840 (p ~ 1e-110). Proving the mechanism is the open line.

Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.

Go deeper

Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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