The ternary Goldbach problem with arithmetic weights attached to two of the variables

TL;DR

This paper investigates a variant of the ternary Goldbach problem involving two primes of specific quadratic forms, deriving an asymptotic formula for the count of solutions.

Contribution

It introduces an asymptotic formula for solutions to the Goldbach problem with primes of the form k^2 + m^2 + 1, a novel approach in additive number theory.

Findings

Derived an asymptotic count for solutions

Extended Goldbach problem to quadratic primes

Provided new insights into prime representations

Abstract

We consider the ternary Goldbach problem with two prime variables of the form $k^2+m^2+1$ and find an asymptotic formula for the number of its solutions.