Counting Matrices of Small Trace

One of the ingredients in a paper that Bram Petri and I submitted in 2016 was a count of integer matrices of determinant 1 with non-negative entries and bounded trace.

Our paper only required an upper bound, but as a number theorist I couldn’t resist the temptation of describing the asymptotics of this function more precisely. In this post we explore do just that, exploring Dirchlet’s hyperbola method along the way.

Read Article →

Maximal Products of a Given Sum

Back in high school, I came across the following contest problem – “What is the largest product of a set of positive integers totaling 20?”

It’s a fun problem, so don’t rush past the spoiler tags too fast. In this post, we’ll give the solution to this problem and discuss a “continuous” version of this question. Namely, what happens when we’re allowed to include real numbers in our product?

Read Article →