On the greatest prime Factor of a polynomial

Abstract

Let ƒ(x) be a non-linear polynomial with rational integer coefficients, and for integral x let P(x) denote the greatest (positive) prime factor of ƒ(x). Pólya (1) has proved that if ƒ(x) is of degree 2 and has distinct roots then P(x)→∞ as x→∞. It is probably well-known that, provided ƒ(x) has distinct roots, this is true whatever the degree of ƒ(x). There does not appear to be a proof of this in the literature, but it is easily deducible from a result of Siegel (2). These results, however, are non-effective, although effective results have been obtained for a number of special polynomials. Chowla (3) has proved that, if ƒ(x) = x2 +1, then P(x)>C log log x, where C is an absolute positive constant. Analogous results have been proved for some polynomials of the form ax2+b and for some of the form ax3 + b by Mahler and Nagell respectively (4).