VÔ CONJECTURE 24(VC24): ON THE VALUES OF THE DIVISOR FUNCTION ON POLYNOMIAL SEQUENCES

Let τ(n) denote the divisor function of the positive integer n. Given a finite set of irreducible polynomials f₁, …, fₖ ∈ ℤ[x] of fixed degrees satisfying a natural local condition, we propose the conjecture that, for every fixed even integer m ≥ 2, there exist infinitely many natural numbers x such that τ(f₁(x)) = τ(f₂(x)) = ⋯ = τ(fₖ(x)) = m.
This conjecture can be regarded as a refinement of classical problems concerning the distribution of arithmetic functions along polynomial sequences. It is supported by probabilistic models of prime factorization as well as by the extended Bateman–Horn philosophy. In this paper, we formulate the conjecture precisely, discuss its motivation, necessary conditions, and its connections with known results and open problems.