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

Teräväinen, J

25 January 2018

Mathematika

## Last Updated:

2021-08-19T01:54:09.457+01:00

64

## DOI:

10.1112/S0025579317000341

20-70

## 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.

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Journal Article