Vô Conjecture 25(VC25):A Bateman–Horn Type Conjecture for Squarefree Almost-Prime Values of Polynomials

This paper proposes a generalization of the Bateman–Horn model aimed at predicting the density of simultaneous occurrences of values having a fixed number of prime factors within a system of arithmetic polynomials. Unlike traditional studies, which primarily focus on counting absolute prime values, this conjecture extends the scope to almost-prime values whose total number of prime factors, counted with multiplicity, attains prescribed values. By combining the local residue density captured by the Bateman–Horn singular series with the global distributional structure of Landau’s formula, we formulate two versions of the conjecture: a weak form asserting that the order of magnitude is correct up to multiplication by absolute constants, and a strong form asserting that the asymptotic formula holds with the exact leading constant.