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 ≡ ± 1 p. The core challenge lies in the "parity problem" of sieve theory, where the classical linear sieve of dimension κ=2 fails to provide a positive lower bound at the sifting limit u ≈ 2. By employing Richert’s weighted sieve and Chen’s switching principle, the paper rigorously demonstrates the existence of almost-prime integers of type P₃ in the interval (N, 6N²]. This result provides an explicit and unconditional realization of sieve methods in the context of forbidden residue classes.