In recent years there has been some spectacular advances in the field of prime numbers. In particular, striking progress has been made on the prime numbers gap conjectures. Classically, the conjecture posits that there are infinitely many pairs of twin prime numbers (i.e., pairs of prime numbers that differs by 2; one such twin pair is 17 and 19). More generally one can ask if there are infinitely many pairs of primes within a fixed gap x.
Inspired by these recent breakthroughs, we review some fascinating properties of prime numbers and of Bungus numbers that can be introduced in mathematics classrooms at almost any level. To make the learning of these notions fun and playful, we introduce three student games. The first one is suitable for most grades, while the other two are suitable for higher grades. We also suggest topics for additional classroom workshops.
Abstract elementary classes (AECs for short) were introduced by Shelah in the seventies to study those classes of structures that can not be axiomatized by a first-order theory. In this talk, we will introduce the basic notions of AECs and showcase them in classes of modules. In particular, we will explore if every AEC of modules with pure embeddings is stable. Using that the class of p-groups with pure embeddings is a stable AEC, we will present a solution to a problem of László Fuchs. For those at CU Boulder, I am assigned to teach Topics in Logic next semester so the talk will also be a preview of the material I am planning to cover in that course.
Stability in abstract elementary classes of modules