Almost-primes in Sun's $x^2+ny^2$ conjecture

TL;DR

This paper investigates Sun's conjecture on representing integers as sums and quadratic forms with primes, using weighted sieve methods to find almost-prime solutions for large integers.

Contribution

It formalizes Sun's conjecture as a sieve problem and applies weighted sieve techniques to obtain partial results with almost-prime solutions.

Findings

Verified conditions for weighted sieve application

Established partial results for large n

Found almost-prime solutions satisfying the conjecture

Abstract

In 2015 Zhi-Wei Sun proposed the conjecture that any integer $n > 1$ admits a partition $n = x + y$ with integers $x, y >0$ such that $x + ny$ and $x^2 + ny^2$ are simultaneously prime. To approach this conjecture we use the method of weighted sieve as developed by Richert, Halberstam, and Diamond. In this article, we first formalize the conjecture into a sieve problem. We verify that the conditions required to use Richert's weighted sieve are satisfied and establish partial results with almost-prime solutions for sufficiently large $n$.