# The Philosophy of Square-Root Cancellation

In this note, I’ll discuss why square-root cancellation is so typical in problems in number theory and give a quick survey of important sums known or widely conjectured to satisfy bounds of this form.

In this note, I’ll discuss why square-root cancellation is so typical in problems in number theory and give a quick survey of important sums known or widely conjectured to satisfy bounds of this form.

Classification theorems of Euler, Lagrange, and Legendre describe the sets of integers that can be written as the sum of 2, 3, and 4 squares. In the last two cases, it follows easily that the density of these sets are 5/6 and 1.

The question of density is not so simple in the case of two squares. In this post, we resolve using an unexpected tool — Dirichlet’s theorem on primes in arithmetic progressions.

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?

In this post, we’ll look at one-player games of limited information (sometimes classified as puzzles, not games) through a topological lens, and create for each game a poset of topologies under which topologically indistinguishable points correspond to outcomes that are indiscernible in a limited-information context. Expanding this dictionary, we’ll describe a topology on the outcome space under which the “safe” or “warranted” extension of one’s limited information relates to the continuity of certain maps.

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?

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.