Arithmetic Progressions in Unique Factorization Domains

Abstract: S. S. Pillai, a contemporary of Ramanujan and perhaps the second best Indian mathematician
at least of that era, proved in 1940 that any sequence of consecutive integers with at most 16 terms
possesses one term that is relatively prime to all the others. Later Brauer and Pillai showed (independently)
that such a result is not true for sequences of length 17 or more. It is not difficult to see that Pillai’s theorem
generalizes from consecutive integers to arithmetic progressions in the sense that any sequence of 16 (or less)
integers in a coprime arithmetic progression (that is, an a.p. for which the first term and common difference is
relatively prime) necessarily contains a term that is relatively prime to all the others.

We will discuss an algebraic extension of this Generalized Pillai Theorem in a significantly wider algebraic context.
Thus, we ask, if a similar result holds for Gaussian integers, or more generally, for unique factorization domains,
or even more generally, for arbitrary integral domains where the notion of GCD (and hence of two elements
being relatively prime) makes sense. A nice example of such GCD domains is the ring of entire functions, thanks
to a result of Helmer, which was also published in 1940.

We will outline a joint work with Samrith Ram that provides some answers to these questions.