Euler-Rabinowitsch polynomials and class number problems revisited

Rabinowitsch Revisited

In the late eighteenth century both Euler and Legendre noticed that n +n+41 is prime for n = 0, 1, 2 . . . 39, and remarked that there are few polynomials with such small degree and coefficients that give such a long string of consecutive prime values. Rabinowitsch, at the 1912 International Congress of Mathematicians [18], showed that n + n + A is prime for n = 0, 1, 2, . . . A − 2 if and only...

A new class of generalized Bernoulli polynomials and Euler polynomials

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava–Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.

The Rabinowitsch-Mollin-Williams Theorem Revisited

We completely classify all polynomials of type x2 x− Δ−1 /4 which are prime or 1 for a range of consecutive integers x ≥ 0, called Rabinowitsch polynomials, where Δ ≡ 1 mod4 with Δ > 1 square-free. This corrects, extends, and completes the results by Byeon and Stark 2002, 2003 via the use of an updated version of what Andrew Granville has dubbed the Rabinowitsch-MollinWilliams Theorem—by Granvi...

Class-number Problems for Cubic Number Fields

The Gauss Class-Number Problems

In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae [Gau86], Gauss put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind’s time, these conjectures have been phrased in the language of quadratic fields. This...