Short Egyptian fractions

What OpenAI claims

Every rational a/b with 1 ≤ a < b is a sum of O(log log b) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions. (Lean)

Paper

1. Short Egyptian fractions

   25 September 2026 · Preprint, not peer reviewed · PDF

   We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number a/b with 1 ≤ a < b is a sum of O(log log b) distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.