{"id":"openai-math/Short-Egyptian-fractions","pdf":"https://github.com/openai/math/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/Short-Egyptian-fractions-September-25-2026","tags":["number-theory","egyptian-fractions","additive-combinatorics"],"score":0,"title":"Short Egyptian fractions","domain":"Number theory","family":"025","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"We prove a conjecture of Erdős: for every sufficiently large integer b , every rational number $`a/b`$ with $`1\\le a\\lt b`$ 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 .","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":true,"family_title":"Short Egyptian fractions"}