The exceptional set in a generalized Goldbach`s problem

TL;DR

This paper investigates the size of the exceptional set in a generalized Goldbach problem and proves the existence of infinitely many integers satisfying a polynomial prime sum condition under certain modular constraints.

Contribution

It computes the exceptional set size and establishes the infinitude of solutions for polynomial prime sums with modular restrictions, extending Goldbach-type results.

Findings

Size of the exceptional set is explicitly computed.

Infinitely many integers satisfy the polynomial prime sum condition.

Results hold under mild modular and polynomial conditions.

Abstract

In this paper, we compute the size of the exceptional set in a generalized Goldbach problem and show that for a given polynomial $f(x) \in \mathbb{Z}[x]$ with a positive leading coefficient, positive integers $A$, $B$, $g$ and $0 \leq i, j < g$, there are infinitely many integers $n$ which satisfy $f(n) = Ap + Bq$ for some primes $p \equiv i, q \equiv j \pmod{g}$ under a mild condition.