Existence of Integers with Few Prime Factors Avoiding Certain Residue Classes Modulo Each Prime

This paper addresses a problem at the intersection of additive combinatorics and analytic number theory: finding integers in a polynomial interval that avoid specific residue classes modulo every prime up to a given bound. The author considers the quadratic sequence $Q(n) = (6n-1)(6n+1)$ and identifies integers $n$ that avoid the classes $6n \equiv \pm 1 \pmod{p}$. The core challenge lies in the "parity problem" of sieve theory, where the classical linear sieve of dimension $\kappa=2$ fails to provide a positive lower bound at the sifting limit $u \approx 2$. By employing Richert’s weighted sieve and Chen’s switching principle, the paper rigorously demonstrates the existence of almost-prime integers of type $P_3$ in the interval $(N, 6N^2]$. This result provides an explicit and unconditional realization of sieve methods in the context of forbidden residue classes.