On the exceptional set for binary Egyptian fractions

Jing Jing Huang, Robert C. Vaughan

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


For fixed integer a ≥ 3, we study the binary Diophantine equation a/n = 1/x + 1/y and in particular the number Ea(N) of n ≤ N for which the equation has no positive integer solutions in x and y. The asymptotic formula Ea(N) ∼ C(α)N(log logN)2m-1-1/(logN)1-1/2m, as N goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of a certain combinatorial property of the underlying group (Z/aZ)* and m depends in a subtle way on the factorization of α.

Original languageEnglish (US)
Pages (from-to)861-874
Number of pages14
JournalBulletin of the London Mathematical Society
Issue number4
StatePublished - Aug 2013

All Science Journal Classification (ASJC) codes

  • General Mathematics


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