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

Part of: Additive number theory; partitions Multiplicative number theory

Published online by Cambridge University Press:  25 January 2018

Joni Teräväinen
Joni Teräväinen*
Affiliation:
Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland email joni.p.teravainen@utu.fi
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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 Matomäki, 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 $\unicode[STIX]{x1D6FC}p~(\text{mod}~1)$, with $\unicode[STIX]{x1D6FC}$ irrational and $p$ running through primes of the form $x^{2}+y^{2}+1$, are distributed rather uniformly.

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 11N36: Applications of sieve methods
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 20 - 70
Copyright
Copyright © University College London 2018 

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