2001/04/12 by Christian Elsholtz · 2 citations
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Combinatorics #Mathematics #Integer (computer science) #Unit (ring theory) #Exponential function #Mathematical analysis
paper · pdf · doi:10.1090/s0002-9947-01-02782-9
openalex publication_date 2001/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
ErdÅs and Straus conjectured that for any positive integer n≥ 2 the equation \frac 4n= \frac 1x + \frac 1y + \frac 1z has a solution in positive integers x, y, and z. Let m > k ≥ 3 and Em,k(N)= | \ n ≤ N | \frac mn = \frac 1t1 + … + \frac 1tk \text has no solution with ti ∈ \mathbb 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 ( -cm,k (log N)^1-\frac 12k-1-1 ) \text with cm,k>0. This improves upon earlier work by Viola (1973) and Shen (1986), and is an âexponential generalizationâ of the work of Vaughan (1970), who considered the case k=3.