Sums of the Form 1/x₁^k + ... + 1/x_n^k Modulo a Prime

2004/03/22 by Ernie Croot, Croot, Ernie
Mathematics · #11P99 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:11P99

paper · pdf · doi:10.48550/arxiv.math/0403360

Light Corrections. The parameter h in the definition of T had to be a lot larger

openalex publication_date 2004/03/22 · arxiv created 2004/10/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for every 0 < ε≤ 1 and integer k≥ 1, there exists an integer n = n(ε,k) so that for all primes p, and integers 0 ≤ a ≤ p-1, there exist integers 1 ≤ x₁ < ... < x_n ≤ p^ε such that a ≡ x₁^-1 + ... + x_n^-1 \pmodp. This extends a result of I. Shparlinski.

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