# Number theory

There have been 4 completed talks, 1 document, and 12 topic suggestions tagged with number theory.

## Completed Talks

### Automatic sequences

Delivered by Laindon Burnett on Wednesday October 25, 2017

This talk will begin with a brief overview behind the theory of words in mathematics as well as the theory of finite automata in theoretical computer science. After this, we will define what an automatic sequence is, prove some fundamental theorems about them, and investigate some of their more intriguing properties. The majority of information presented comes from the text “Automatic Sequences: Theory, Applications, Generalisations” by Jean-Paul Allouche and the University of Waterloo's own Jeffrey Shallit, from the department of Computer Science.

The speaker has provided a PDF document covering the same content as this talk.

### $p$-adic Numbers

Delivered by Akshay Tiwary on Friday February 3, 2017

In this talk we will discover an alternate way to define the "size" of a ration al number. We will define the p-adic absolute value and see that this absolute value is Non-archimedean and that this fact leads to a a geometry that is very different from the geometry of the Real numbers (every p-adic triangle is isosc eles!). In addition I will show you some fun ideas from p-adic numbers (which I will denote by $\mathbb{Q}_p$ like the fact that $\sum_{n = 1}^{\infty} 3^n$ converges and how every element of $\mathbb{Q}_p$ has a base $p$ expansion. This is meant to be a very leisurely talk with almost no prerequisites and it should be fun so please come for it!

### Diophantine Approximation and the Discovery of Transcendental Numbers

Delivered by Anton Mosunov on Friday January 20, 2017

In 1844, Joseph Liouville discovered transcendental numbers — those numbers that are not roots of polynomials with rational coefficients. Nowadays, we see Liouville’s discovery as the foundation of Diophantine approximation, a fascinating subarea of number theory, which studies how well algebraic numbers can be approximated by the rationals. In my talk, I will prove the classical result of Liouville and explain further advances in the area, such as Thue’s theorem and the celebrated theorem of Roth, which enabled its discoverer to receive the Fields medal in 1958.

### Basic Elliptic Curves

Delivered by Akshay Tiwary on Friday November 18, 2016

Elliptic Curves lie in the intersection of Algebra, Geometry, Number Theory and Complex Analysis. While my talk won't require any experience with complex analysis or algebraic geometry, I hope to expose you to this active area of research. Although I could tell you that the meat of Wiles' proof of Fermat's Last Theorem involved proving a special case conjecture previously known as the Taniyama Shimura conjecture involving elliptic curves, or that the Birch and Swinnerton-Dyer conjecture is an open millenium prize problem that talks about the relation between the arithmetic behaviour of elliptic curves and the analytic behaviour of an L-function, or that elliptic curves over finite fields is so useful for cryptography that there was a memo that recommended elliptic curves for federal government use in 1999, I don't have to because you're a math student who will attend this talk without needing to be wowed with cool applications, right?

## Documents

This is a reference document on Quadratic Reciprocity by Fengyang Wang. It covers a subset of the material of PMATH 340 or PMATH 440.

## Talk Suggestions

### Almost Integers and Pisot Numbers

Sometimes some non-integers like $e^{π} - π = 19.9990999791...$ are curiously close to integers. While often this is just a numerical coincidence, sometimes there is a clear mathematical reason for certain numbers to be almost integers. Pisot numbers, which are special types of algebraic integers, provide a systematic means to construct almost integers that approximate integers at an exponential rate. Of course, Pisot numbers are interesting on their own right and an interested speaker can also venture into the connections to Beta expansions (see the 2nd and 3rd references below), or fractals (see the 4th reference below).

Required Background: Basic analysis at the level of 147 and 138, algebra at the level of 145. Any other background depends on the direction the speaker takes with this topic.

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### Continued Fractions and Hyperbolic Geometry

This topic studies the beautiful connection between continued fractions and the famous Farey tessekation of the hyperbolic plane.

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### Costas Arrays

Permutation matrices for which all $\frac{n(n-1)}{2}$ displacements between all pairs of 1s are distinct are known as Costas arrays, and these have important applications in sonar. The Welch construction of Costas arrays uses concepts from number theory, whereas the Lempel-Golomb construction uses concepts from field theory.

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### Dirichlet's Theorem on Primes in Arithmetic progression

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### Distribution of Primes Modulo p

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### Lattices, Linear Codes, and Invariants

This topic studies sphere packing and the connection to lattices and linear codes.

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### Markov Equation and Irrational Numbers

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### Patterns in Primes

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### The Joy of Factoring

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### The Riemann Hypothesis

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