A1 Refereed original research article in a scientific journal

THE GOLDBACH PROBLEM FOR PRIMES THAT ARE SUMS OF TWO SQUARES PLUS ONE




AuthorsTeravainen J

PublisherLONDON MATH SOC

Publication year2018

JournalMathematika

Journal name in sourceMATHEMATIKA

Journal acronymMATHEMATIKA

Volume64

Issue1

First page 20

Last page70

Number of pages51

ISSN0025-5793

eISSN2041-7942

DOIhttps://doi.org/10.1112/S0025579317000341(external)


Abstract
We study the Goldbach problem for primes represented by the polynomial x(2) + y(2) + 1. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers n satisfying certain necessary local conditions are representable as the sum of two primes of the form x(2) + y(2) + 1. This improves a result of Matomiki, which tells us that almost all even n satisfying a local condition are the sum of one prime of the form x(2) + y(2) + 1 and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd n is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form x(2) + y(2) + 1 contain infinitely many three-term arithmetic progressions, and that the numbers up (mod 1), with alpha irrational and p running through primes of the form x(2) + y(2) + 1, are distributed rather uniformly.



Last updated on 2024-26-11 at 18:52