The Goldbach Problem for Primes That Are Sums of Two Squares Plus One

TL;DR

This paper advances the understanding of the Goldbach problem for primes of the form x^2 + y^2 + 1, proving that almost all even integers satisfying certain conditions are sums of two such primes and solving the ternary problem for large odd integers.

Contribution

It proves the representation of almost all relevant even integers as sums of two primes of the form x^2 + y^2 + 1 and establishes the ternary Goldbach conjecture for these primes.

Findings

Almost all even integers meeting local conditions are sums of two primes of the form x^2 + y^2 + 1.

Every large odd integer can be expressed as a sum of three such primes.

Primes of the form x^2 + y^2 + 1 contain infinitely many three-term arithmetic progressions.

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 Matom\"aki, which tells 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 $\alpha p \pmod 1$ with $\alpha$ irrational and $p$ running through primes of the form $x^2+y^2+1$, are distributed rather uniformly.