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

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 $n$.

💡 Research Summary

The paper tackles Zhi‑Wei Sun’s 2015 conjecture that every integer (n>1) can be written as a sum (n=x+y) with positive integers (x,y) such that both (x+ny) and (x^{2}+ny^{2}) are prime. Since proving the simultaneous primality of these two expressions appears out of reach with current technology, the authors relax the problem to an “almost‑prime” version, i.e., they seek (y) for which the numbers have only a bounded number of prime factors (counted with multiplicity).

The authors reformulate the conjecture in terms of two polynomials in the variable (y): \