Number Theory seminar

Michael Filaseta, University of South Carolina

“49598666989151226098104244512918”

If p is a prime with decimal representation dndn−1 . . . d1d0, then a theorem of A. Cohn implies that the polynomial f(x) = dnxn + dn−1xn−1 + · · · + d1x + d0 is irreducible. One can view this result as following from the fact that if g(x) ∈ Z[x] with g(0) = 1, then g(x) has a root in the disk D={z∈C:|z|≤1}.  On the other hand, that such ag(x) has a root in D has little to do with g(x) having integer coefficients. In this talk, we discuss a perhaps surprising result about the location of a zero of such a g(x) that makes use of its coefficients being in Z and discuss the implications this has on generalizations of Cohn’s theorem. A variety of open problems will be presented. This research is joint work with a now former student, Sam Gross.

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