2013/02/01 by Christian Elsholtz, Terence Tao · 2 citations
Mathematics · #Analytic Number Theory Research #Mathematical Dynamics and Fractals #Algebraic Geometry and Number Theory #Mathematics #Combinatorics #Integer (computer science) #Diophantine equation #Conjecture #Prime factor #Prime number #Prime (order theory)
paper · pdf · doi:10.1017/s1446788712000468
openalex publication_date 2013/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Abstract For any positive integer n , let f(n) denote the number of solutions to the Diophantine equation \begineqnarray*(4)/(n) = (1)/(x) + (1)/(y) + (1)/(z)\endeqnarray* with x, y, z positive integers. The Erdős–Straus conjecture asserts that f(n)\gt 0 for every n≥ 2 . In this paper we obtain a number of upper and lower bounds for f(n) or f(p) for typical values of natural numbers n and primes p . For instance, we establish that \begineqnarray*N\hspace0.167em \mathoplog \nolimits 2 N≪ ∑ p≤ Nf(p)≪ N\hspace0.167em \mathoplog \nolimits 2 Nlog log N.\endeqnarray* These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.