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

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 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 $\alpha p(\mathrm{mod}1)$ with $\alpha$ irrational and $p$ running through primes of the form $x^2+y^2+1$, are distributed rather uniformly.

49 pages; Referee comments incorporated; To appear in Mathematika

References in corpus (1)

• Vinogradov's three primes theorem with almost twin primes

Cited by in corpus (5)

• Vinogradov's Theorem with Fouvry-Iwaniec Primes
• Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory
• A transference principle for systems of linear equations, and applications to almost twin primes
• The ternary Goldbach problem with prime numbers of a mixed type
• The ternary Goldbach problem with a prime with a missing digit and primes of special types