Square-free values of reducible polynomials

TL;DR

This paper determines the conditions under which reducible polynomials with integer coefficients, no fixed prime divisor, and factors of degree at most 3, take infinitely many values that are products of a limited number of distinct primes.

Contribution

It provides a comprehensive calculation of admissible values of r for which such polynomials take infinitely many r-almost prime values.

Findings

Identifies the specific values of r for which the polynomial values are r-almost primes.

Establishes conditions on polynomial reducibility and degree for infinite r-almost prime values.

Extends previous results to reducible polynomials with low-degree factors.

Abstract

We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct primes.