Sums of k unit fractions

Christian Elsholtz*

*Corresponding author for this work

    Research output: Contribution to journalArticle

    Abstract

    Erdõs and Straus conjectured that for any positive integer n ≥ 2 the equation 4/n = 1/x + 1/y + 1/x has a solution in positive integers z, y, and z. Let m > k ≥ 3 and Em,k(N) =| {n ≤ N | m/n = 1/t1 + ⋯ + 1/tkhas no solution with ti ∈ ℕ} | . We show that parametric solutions can be used to find upper bounds on Em,k(N) where the number of parameters increases exponentially with k. This enables us to prove Em,k(N) ≪ N exp(-cn,k(log N)1-1/2k-1-1) with cm,k > 0. This improves upon earlier work by Viola (1973) and Shen (1986), and is an "exponential generalization" of the work of Vaughan (1970), who considered the case k = 3.

    Original languageEnglish
    Pages (from-to)3209-3227
    Number of pages19
    JournalTransactions of the American Mathematical Society
    Volume353
    Issue number8
    Publication statusPublished - 2001

    ASJC Scopus subject areas

    • Mathematics(all)

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