Square-free values of reducible polynomials

Square-free values of reducible polynomials, Discrete Analysis, 2016:8, 18 pp.

When does a polynomial with integer coefficients take infinitely many prime values? An obvious necessary condition is that it should be irreducible, and another is that there should be no prime $p$ such that the values taken by the polynomial are all multiples of $p$. (The second condition rules out the irreducible polynomial $x^3-x+3$, for example, all of whose values are multiples of 3.) It is conjectured that these conditions are sufficient, but that problem is wide open: Dirichlet’s theorem on primes in arithmetic progression implies the conjecture for linear polynomials, but even for the simplest quadratics such as $x^2+1$ the answer is not known.

Some progress has been made on weaker questions, however. For example, if the polynomial is quadratic, then infinitely often it is the product of two primes, and for higher degree $d$ it is infinitely often a product of $d+1$ primes.

A significant further weakening is to ask for square-free values instead of primes. Erdős proved that polynomials that have degree at most 3 and satisfy the conditions above take infinitely many square-free values, but the answer is unknown for a single such polynomial of higher degree, except that if one accepts the abc-conjecture, then one obtains a positive answer for all such polynomials.

This paper looks at a kind of combination of these two problems: now we would like to find values of polynomials that are both “almost primes”, meaning that they have few prime factors, and square free, meaning that those prime factors must be distinct. They also weaken the hypothesis, assuming not that their polynomials are irreducible, but merely that they are a product of at most $\kappa$ irreducible factors. In the light of our knowledge about polynomials taking square-free values, they must also assume that these factors have degree at most 3.

Under these assumptions, they prove that the polynomial must infinitely often take values that are a product of at most $r$ distinct primes, where $r$ depends on $\kappa$ only. The proof uses sieve methods, and also requires significant computer calculations. The authors have made the results of these calculations available online, as well as their source code.