Cohen-Kaplansky domains and the Goldbach conjecture

Coykendall, Jim; Spicer, Chris

doi:10.1090/S0002-9939-2011-11086-4

Cohen-Kaplansky domains and the Goldbach conjecture
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by Jim Coykendall and Chris Spicer PDF

Proc. Amer. Math. Soc. 140 (2012), 2227-2233 Request permission

Abstract:

A Cohen-Kaplansky domain is an atomic domain with only a finite number of irreducibles. In this paper, we show that localizations of certain orders of rings of integers are necessarily CK-domains, and then prove there exists a closed form formula for the number of irreducible elements in several different cases of these types of rings. Modulo a variant of the Goldbach Conjecture, this construction allows us to answer a question posed by Cohen and Kaplansky over 60 years ago regarding the construction of a CK-domain containing $n$ nonprime irreducible elements for every positive integer $n$.

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