Almost-primes in Sun's $x^2+ny^2$ conjecture
Songlin Han, Jinbo Yu
TL;DR
This work addresses Sun's conjecture that every integer $n>1$ admits a partition $n=x+y$ with $x+ny$ and $x^2+ny^2$ simultaneously prime, by formulating it as a weighted sieve problem. The authors deploy a multi-dimensional, Diamond–Halberstam–Richert type weighted sieve to derive almost-prime results for large $n$, proving two concrete bounds: there exists $0<y<n$ with $ Omega\big(n+(n-1)y\big)\le 3$ and there exists $0<y<n$ with $ Omega\big((n+1)y^2-2ny+n^2\big)\le 4$. These results rely on establishing sieve density functions, local/global densities, and the key inequality for the weighted sifting function $W(\mathcal{A},u,\lambda)$, together with careful handling of error terms. The findings demonstrate the effectiveness of advanced sieve methods in producing almost-prime representations related to Sun's conjecture and provide a framework for potential refinements toward prime-level analogues.
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 $n$.