Smooth values of polynomials

Jonathan Bober (University of Bristol)

Frank Adams 1,

It is conjectured that for any nonzero polynomial f(x) with integer coefficients and any epsilon > 0, there should be infinitely many integers n such that all of the prime factors of f(n) are < n^epsilon. Aside from some polynomials with a special form, this conjecture has remained unproven for polynomials of degree > 1.  I will describe recent progress in degree 2 and related questions about when there exist polynomials g(x) such that f(g(x)) is a smooth polynomial. Despite this seemingly being a question in the realm of analytic number theory, most of the methods will involve little more than undergraduate level algebra. (This talk is based on joint work with Dan Fretwell, Greg Martin, and Trevor Wooley.)
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