Graduate Number Theory Seminar
Robert Krzyzanowski
UIC
Prime-generating polynomials
Abstract: In 1772, Euler noticed the polynomial $x^2 + x + 41$ produces forty consecutive primes for $0 <= x <= 39$. We will show that this fact is actually equivalent to $Q(\sqrt{-d})$
with $d = 163 = 1 - 4*41$ having class number 1. Further, $x^2 + x + m$ is such a prime-generating polynomial if and only if $Q(\sqrt{1-4m})$ has class number 1. This question
can be generalized to real quadratic fields by letting $m$ be negative. There are then only 14 such polynomials, called Rabinowitsch polynomials.
Wednesday February 17, 2010 at 3:00 PM in SEO 427