The application

During this year’s MOP, we used the following procedure to divide some of our students into two classes:

Let $p = 7075374838595186541578161$ be prime. Take the letters in your name as it appears on the roster, convert them with A1Z26 and take the sum of cubes to get a number $s$. For example, EVANCHEN corresponds to $s = 5^3 + 22^3 + \dots + 14^3 = 16926$. Then you’re in Red 1 (room A155) if $s$ is a quadratic residue modulo $p$, and Red 2 (room A133) otherwise.

The students were understandably a bit confused why the prime was chosen. It turned out to be a prank: if you ran the calculation on the 30-ish students in this class, it was …

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’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 was my final paper for 18.099, seminar in discrete analysis, jointly with Sammy Luo and Ryan Alweiss.

We prove that every sufficiently large odd integer can be written as the sum of three primes, conditioned on a strong form of the prime number theorem.

1. Introduction

In this paper, we prove the following result:

Theorem 1 (Vinogradov)

Every sufficiently large odd integer $N$ is the sum of three prime numbers.

In fact, the following result is also true, called the “weak Goldbach conjecture”.

Theorem 2 (Weak Goldbach conjecture)

Every odd integer $N \ge 7$ is the sum of three prime numbers.

The proof of Vinogradov’s theorem becomes significantly simpler if one assumes the generalized Riemann hypothesis; this allows one to use a strong form of the prime number theorem (Theorem 9). This conditional proof was given by Hardy and Littlewood in …

This is a transcript of a talk I gave as part of MIT’s 18.434 class, the “Seminar in Theoretical Computer Science” as part of MIT’s communication requirement. (Insert snarky comment about MIT’s CI-* requirements here.) It probably would have made a nice math circle talk for high schoolers but I felt somewhat awkward having to present it to a bunch of students who were clearly older than me.

1. Preliminaries

1.1. Modular arithmetic

In middle school you might have encountered questions such as

Exercise 1. What is $3^{2016} \pmod{10}$?

You could answer such questions by listing out $3^n$ for small $n$ and then finding a pattern, in this case of period $4$. However, for large moduli this “brute-force” approach can be time-consuming.

Fortunately, it …

I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define $L$-series for non-abelian extensions. But for them to agree with the $L$-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I called it the General Reciprocity Law and tried to prove it but couldn’t, even after many tries. Then I showed it to the other number theorists, but they all laughed at it, and I remember Hasse in particular telling me it couldn’t possibly be true.

Still, I kept at it, but nothing I tried worked. Not a week went by — for three years! — that I did not try to prove the Reciprocity Law. It was discouraging, and meanwhile I turned to other things. Then one afternoon I had nothing …

There are some notes on valuations from the first lecture of Math 223a at Harvard.

1. Valuations

Let $k$ be a field.

Definition 1. A valuation $$\left\lvert - \right\rvert : k \rightarrow \mathbb R_{\ge 0}$$ is a function obeying the axioms

• $\left\lvert \alpha \right\rvert = 0 \iff \alpha = 0$.
• $\left\lvert \alpha\beta \right\rvert = \left\lvert \alpha \right\rvert \left\lvert \beta \right\rvert$.
• Most importantly: there should exist a real constant $C$, such that $\left\lvert 1+\alpha \right\rvert < C$ whenever $\left\lvert \alpha \right\rvert \le 1$.

The third property is the interesting one. Note in particular it can be rewritten as $\mid a+b\mid <C\max \{\mid a\mid ,\mid \mathrm{b\; \dots }$

Here I talk about my first project at the Emory REU. Prerequisites for this post: some familiarity with number fields.

1. Motivation: Arithmetic Progressions

Given a property $P$ about primes, there’s two questions we can ask:

1. How many primes $\le x$ are there with this property?
2. What’s the least prime with this property?

As an example, consider an arithmetic progression $a$, $a+d$, …, with $a < d$ and $\gcd(a,d) = 1$. The strong form of Dirichlet’s Theorem tells us that basically, the number of primes $\equiv a \pmod d$ is $\frac 1d$ the total number of primes. Moreover, the celebrated Linnik’s Theorem tells us that the first prime is $O({\mathrm{d\; \dots }}^{}$

In this post I will sketch a proof Dirichlet Theorem’s in the following form:

Theorem 1 (Dirichlet’s Theorem on Arithmetic Progression)

Let $$\psi(x;q,a) = \sum_{\substack{n \le x \\ n \equiv a \mod q}} \Lambda(n).$$ Let $N$ be a positive constant. Then for some constant $C(N) > 0$ depending on $N$, we have for any $q$ such that $q \le (\log x)^N$ we have $$\psi(x;q,a) = \frac{1}{\phi(q)} x + O\left( x\exp\left(-C(N) \sqrt{\log …$$