The ternary Goldbach problem with prime numbers of a mixed type

TL;DR

This paper proves that sufficiently large odd integers can be expressed as the sum of three primes, with two primes of special forms, advancing understanding of the Goldbach problem for mixed prime types.

Contribution

It establishes the representation of large odd integers as sums of three primes with specific algebraic and fractional form constraints, extending classical Goldbach results.

Findings

Every large odd integer N can be written as p1 + p2 + p3

p1 is of the form x^2 + y^2 + 1

p2 is of the form [n^c]

Abstract

In the present paper we prove that every sufficiently large odd integer $N$ can be represented in the form \begin{equation*} N=p_1+p_2+p_3\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$, $p_2=[n^c]$.