# Units Groups and the Infinitude of Primes (Part II)

This post serves as a follow-up article to “Units Groups and the Infinitude of Primes”, and looks at generalizations of those results to a wide class of commutative rings.

This post serves as a follow-up article to “Units Groups and the Infinitude of Primes”, and looks at generalizations of those results to a wide class of commutative rings.

This post concerns generalizations of Euclid’s proof of the infinitude of primes to number fields, where it can be viewed as a relationship between the size of the units group and the number of prime ideals in a ring.

The full list of imaginary quadratic fields with unique factorization has been determined, but it remains unknown whether or not infinitely many real quadratic fields are UFDs. Here, we develop a simple criterion for disproving unique factorization based on continued fractions and use this to give upper bounds.

The Classification of Finite Simple Groups proves in particular that the collection of orders of finite simple groups has asymptotic density 0. What can we prove if we’re not willing to work that hard?

Landau considered the following question – What is the maximal order of an element in the permutation group on k letters? We’ll prove some elementary bounds and deduce an asymptotic using the PNT.

Does there exist a pair of loaded six-sided dice such that the probability of rolling any dice sum {2,..12} is equally likely? We’ll show how this and other related questions about dice sums can be analyzed using cyclotomic polynomials.