Two topics in combinatorial number theory

Abstract: The first topic: In a paper with Erd\H os from 40 years ago, we considered the set of residues $a \bmod n$ where $a^{n-1} \equiv 1 \pmod n$. If $n$ is composite, these are the bases for which $n$ is a pseudoprime. Recently, Lenstra asked me about the set of residues $a \bmod n$ where $a^n \equiv 1 \pmod n$, which is related to a problem he is working on about conditions that ensure a ring is commutative. Some of the methods from the old paper were useful in the new problem, but not all. I will discuss the more general problem of subgroups of the multiplicative group mod $n$. The second topic: I will discuss some old and new problems on coprime matchings: These are perfect matchings between two equally numerous sets of integers, where each matched pair is relatively prime. Some examples: Given two intervals of $n$ consecutive integers is there a coprime matching between them? If both intervals are $\{1,2,\dots,n\}$, how many such matchings are there? For a positive integer $n$, is there a coprime matching between the set $D(n)$ of divisors of $n$ and an interval of $D(n)$ consecutive integers? This last problem reflects joint work with Nathan McNew.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)