In the previous post we defined $p$-adic numbers. This post will state (mostly without proof) some more surprising results about continuous functions $f \colon \mathbb Z_p \rightarrow \mathbb Q_p$. Then we give the famous proof of the Skolem-Mahler-Lech theorem using $p$-adic analysis.

1. Digression on $\mathbb C_p$

Before I go on, I want to mention that $\mathbb Q_p$ is not algebraically closed. So, we can take its algebraic closure $\overline{\mathbb Q_p}$ — but this field is now no longer complete (in the topological sense). However, we can then take the completion of this space to obtain $\mathbb C_p$. In general, completing an algebraically closed field remains algebraically closed, and so there is a larger space $\mathbb …$

I think this post is more than two years late in coming, but anywhow…

This post introduces the $p$-adic integers $\mathbb Z_p$, and the $p$-adic numbers $\mathbb Q_p$. The one-sentence description is that these are “integers/rationals carrying full mod $p^e$ information” (and only that information).

The first four sections will cover the founding definitions culminating in a short solution to a USA TST problem.

In this whole post, $p$ is always a prime. Much of this is based off of Chapter 3A from Straight from the Book.

1. Motivation

Before really telling you what $\mathbb Z_p$ and $\mathbb Q_p$ are, let me tell you what you might expect them to do.

In elementary/olympiad number theory, we’re already well-familiar …

I’ve added a new Euclidean geometry handout, Constructing Diagrams, to my webpage.

Some of the stuff covered in this handout:

• Advice for constructing the triangle centers (hint: circumcenter goes first)
• An example of how to rearrange the conditions of a problem and draw a diagram out-of-order
• Some mechanical suggestions such as dealing with phantom points
• Some examples of computer-generated figures

Enjoy.

1. Synopsis

One of the major headaches of using complex numbers in olympiad geometry problems is dealing with square roots. In particular, it is nontrivial to express the incenter of a triangle inscribed in the unit circle in terms of its vertices.

The following lemma is the standard way to set up the arc midpoints of a triangle. It appears for example as part (a) of Lemma 6.23.

Theorem 1 (Arc midpoint setup for a triangle)

Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $M_A$, $M_B$, $M_C$ denote the arc midpoints of $\widehat{BC}$ opposite $A$, $\widehat{CA}$ opposite $B$, $\widehat{AB}$ opposite $C$.

Suppose we view $\Gamma …$

Median Putnam contestants, willing to devote one of the last Saturdays before final exams to a math test, are likely to receive an advanced degree in the sciences. It is counterproductive on many levels to leave them feeling like total idiots.

— Bruce Reznick, “Some Thoughts on Writing for the Putnam”

Last February I made a big public apology for having caused one of the biggest scoring errors in HMMT history, causing a lot of changes to the list of top individual students. Pleasantly, I got some nice emails from coaches who reminded me that most students and teams do not place highly in the tournament, and at the end of the day the most important thing is that the contestants enjoyed the tournament.

So now I decided I have to apologize for 2016, too.

The story this time is that I inadvertently sent over 100 students home having solved two …

I recently had a combinatorics paper appear in the EJC. In this post I want to brag a bit by telling the “story” of this paper: what motivated it, how I found the conjecture that I originally did, and the process that eventually led me to the proof, and so on.

This work was part of the Duluth REU 2017, and I thank Joe Gallian for suggesting the problem.

1. Background

Let me begin by formulating the problem as it was given to me. First, here is the definition and notation for a “block-ascending” permutation.

Definition 1. For nonnegative integers $a_1$, …, $a_n$ an $(a_1, \dots, a_n)$-ascending permutation is a permutation on $\{1, 2, \dots, a_1 + \dots + a_n\}$ whose descent set is …

I wanted to quickly write this proof up, complete with pictures, so that I won’t forget it again. In this post I’ll give a combinatorial proof (due to Joyal) of the following:

Theorem 1 (Cayley’s Formula)

The number of trees on $n$ labeled vertices is $n^{n-2}$.

Proof: We are going to construct a bijection between

• Functions $\{1, 2, \dots, n\} \rightarrow \{1, 2, \dots, n\}$ (of which there are $n^n$) and
• Trees on $\{1, 2, \dots, n\}$ with two distinguished nodes $A$ and $B$ (possibly $A=B$).

This will imply the answer.

Let’s look at the first piece of data. We can visualize it as $\mathrm{n\; \dots }$

I’m reading through Primes of the Form $x^2+ny^2$, by David Cox (it’s good!). Here are the high-level notes I took on the first chapter, which is about the theory of quadratic forms.

(Meta point re blog: I’m probably going to start posting more and more of these more high-level notes/sketches on this blog on topics that I’ve been just learning. Up til now I’ve been mostly only posting things that I understand well and for which I have a very polished exposition. But the perfect is the enemy of the good here; given that I’m taking these notes for my own sake, I may as well share them to help others.)

1. Overview

Definition 1. For us a quadratic form is a polynomial $Q=Q(x,y)=a{x}^{\mathrm{2\; \dots }}$

(This is a bit of a follow-up to the solution reading post last month. Spoiler warnings: USAMO 2014/6, USAMO 2012/2, TSTST 2016/4, and hints for ELMO 2013/1, IMO 2016/2.)

I want to say a little about the process which I use to design my olympiad handouts and classes these days (and thus by extension the way I personally think about problems). The short summary is that my teaching style is centered around showing connections and recurring themes between problems.

Now let me explain this in more detail.

1. Main ideas

Solutions to olympiad problems can look quite different from one another at a surface level, but typically they center around one or two main ideas, as I describe in my post on reading solutions. Because details are easy to work out once you have the main idea, as far as learning is concerned you can …

In this post we’ll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation.

Let $f \colon U \rightarrow \mathbb C$ be a holomorphic function. A holomorphic $n$-th root of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = g(z)^n$ for all $z \in U$. A logarithm of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = e^{g(z)}$ for all $z \in U$.

The main question …