On the Waring-Goldbach problem with almost equal summands

2019/03/05 by Salmensuu, Juho
#11B30 #11P05 #11P32 #11P55 #37A45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1903.01824

Abstract

We use transference principle to show that whenever s is suitably large depending on k ≥ 2, every sufficiently large natural number n satisfying some congruence conditions can be written in the form n = p₁^k + … + p_s^k, where p₁, …, p_s ∈ [x-x^θ, x + x^θ] are primes, x = (n/s)^1/k and θ= 0.525 + ε. We also improve known results for θ when k ≥ 2 and s ≥ k² + k + 1. For example when k ≥ 4 and s ≥ k² + k + 1 we have θ= 0.55 + ε. All previously known results on the problem had θ> 3/4.

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