In the abelian group $\Z/n$, the order of $m\in \Z/n$ can be calculated via the formula $n/{\rm gcd}(m,n)$. This number is just the smallest number you have to multiply $m$ by in order to get a multiple of $n$. So when $n = p$ is a prime, every we see that every nonzero element of $\Z/p$ has order $p$.

However, every nonzero element of $\Z/p$ is prime to $p$, and the nonzero elements of $\Z/p$ form the abelian multiplicative group $\Z/p^\times$ of invertible elements. Then the order of an element $m\in \Z/p^\times$ is a little more tricky to figure out, but we know by Fermat’s Little Theorem that it will be a factor of $p-1$.

Example. Consider the prime $p=11$. Then we know that $3^{10} = 1$ by Fermat’s Little Theorem. However, $3^5 = 1$ too. In fact the order of $~3$ is $~5$.

This example has a curious property that you may not know about. It is true that $3^5 = 1$ in $\Z/11$. But did you know that $3^5 = 1$ in $\Z/121$? What?! This leads to a natural question:

Question. Which primes $p$ have the property that there exists an element $m\in \Z/p^\times, m\not=1$ such that the order of $m$ in $\Z/p^\times$ is the same as the order of $m$ in ${\Z/p^2}^\times$?

Not all primes have this property. Consider $p = 5$. Then $3^4 = 1$ in $\Z/5$. However, $3^4 = 6$ in $\Z/25$. In fact, none of the elements $2,3,4$ have the same order in both $\Z/5^\times$ and $\Z/25^\times$. Here are some primes that do have such elements:

11, 29, 37, 43, 59, 71, 79, 97, 103, 109, 113, 127, …

So, there are quite a few of them and maybe there are infinitely many. One can go further with this question: what about primes for which there are non-trivial elements in $\Z/p^\times$ with the same order in $\Z/p^\times, {\Z/p^2}^\times,$ and ${\Z/p^3}^\times$? There are fewer of these. But they exist. For example, the identity
$$68^{112} = 1$$
Holds in $\Z/113^k$ for $k=1,2,3$ (but not for $k=4$). This is the only example I know of and maybe it is the only possible example? But I probably just haven’t looked far enough. I wrote the original program to find it in Python but I plan to rewrite it in C with the gmp library to expand the search.

Briefly, the Selberg class is a set of functions $F:\C\to\C$ such that $f(s)$ can be written as a Dirichlet series for $\Re(s) > 1$ and that satisfies a form of analytic continuation, a functional equation, a Ramanujan hypothesis bound on coefficients of the Dirichlet series, and an Euler product formula.

Andrew Booker in [1] has extended the Selberg class in a different way, in the notion of an $L$-datum. In this post, we’ll state Booker’s definition of an $L$-datum, state his converse theorem, and explain his corollary that the completed $L$-function of a unitary cuspidal automorphic representation of $\GL_3(\A_\Q)$ has infinitely many zeroes of odd order.
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Posted by Jason Polak on 25. August 2015 · Write a comment · Categories: algebraic-geometry, number-theory · Tags: ,

In the Arthur-Selberg trace formula and other formulas, one encounters so-called ‘orbital integrals’. These integrals might appear forbidding and abstract at first, but actually they are quite concrete objects. In this post we’ll look at an example that should make orbital integrals seem more friendly and approachable. Let $k = \mathbb{F}_q$ be a finite field and let $F = k( (t))$ be the Laurent series field over $k$. We will denote the ring of integers of $F$ by $\mathfrak{o} := k[ [t]]$ and the valuation $v:F^\times\to \mathbb{Z}$ is normalised so that $v(t) = 1$.

Let $G$ be a reductive algebraic group over $\mathfrak{o}$. Orbital integrals are defined with respect to some $\gamma\in G(F)$. Often, $\gamma$ is semisimple, and regular in the sense that the orbit $G\cdot\gamma$ has maximal dimension. One then defines for a compactly supported smooth function $f:G(F)\to \mathbb{C}$ the orbital integral
$$
\Ocl_\gamma(f) = \int_{I_\gamma(F)\backslash G(F)} f(g^{-1}\gamma g) \frac{dg}{dg_\gamma}.
$$
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Posted by Jason Polak on 14. April 2014 · Write a comment · Categories: number-theory

Here’s a cool picture I made with Sage and R of class numbers of $\mathbb{Q}(\sqrt{-D})$ where $D$ is squarefree. It consists of the first five thousand such $D$ (click image to enlarge):

Rplot001

The even class numbers are shown in red +-signs and the odd class numbers are shown as blue disks. If you look carefully at the bottom left-hand corner where the points start, you’ll see a small conglomeration of blue dots that consist of the only nine imaginary quadratic fields that have class number one.

Posted by Jason Polak on 08. June 2013 · Write a comment · Categories: number-theory · Tags: , ,

frame8

Have you ever tried to visualise the graph of a complex function $ f:\mathbb{C}\to\mathbb{C}$? The problem with complex functions is that usually we graph a complex number as an ordered pair $ (x,y)$ on a Euclidean plane, which corresponds to $ z = x + iy$. Unfortunately, this means that if we want to graph complex functions as we do real functions, we need to draw the graph in four-dimensional space! Some people have actually claimed the ability to visualise this, but I do not!

However, if we use time as a dimension, we could represent four dimensions as a moving three-dimensional image in time, like a movie. Sometimes, it’s hard to draw three dimensions in two dimensions, though we don’t actually lose too much because we can only see a two-dimensional picture of a three-dimensional scene at any given time.

There are a few animations in this post; they may take a few seconds to load, or a few minutes on a slower connection.
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Mostly to take a break from marking exams, I thought I’d start a new recurring series here about mathematics papers and books that I find, both new and old. The “new” will consist mainly of preprints that look interesting (to encourage me to browse the arXiv) and the “old” will consists of papers I will likely read (to encourage me to read, or at least skim, more papers).

New Preprints

  1. Clark Barwick, “On the algebraic K-theory of higher categories”: For a while I’ve wanted to learn a bit about higher category theory, but I’ve not yet found any application or motivation for it in something I’m already really interested in (including the stuff here). Perhaps this paper by Barwick will change my mind: algebraic K-theory may be best viewed as a functor with a universal property that generalises the well-known universal properties known for the classical K groups.
  2. Booker, Hiary, and Keating, Detecting squarefree numbers”: Under the Generalised Riemann Hypothesis the authors propose an algorithm to test whether an integer is squarefree, without needing the number’s factorisation.
  3. Shalit, “A sneaky proof of the maximum modulus principle”: This is a proof of the maximum modulus principle in complex analysis, now with even less complex analysis! This paper also appeared in the American Mathematical Monthly and the author wrote a blog post about it as well.

View the whole post to reveal the hidden classic:
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This is a fairly recent picture of the McGill campus:

mcgillcampus

However, soon the spell of unbearable heat will dawn on the city and there will be plenty of fun things to do outside. Despite the hot sun we shouldn’t neglect the indoor activities, such as the many awesome conferences and workshops that will be going on. This post is a list of a few of them that I think would be interesting to people who like algebra and number theory. They are listed in chronological order.
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Posted by Jason Polak on 29. January 2013 · Write a comment · Categories: number-theory · Tags: , ,

The 3n+1 conjecture (a.k.a. the Collatz conjecture) is easy to state, unsolved, probably difficult to prove if true, and can provide us with some pretty pictures. I’m only going to state it and show some pretty pictures.

First, what’s this conjecture? It’s about a simple algorithm: choose a positive integer $ n$. If $ n=1$, don’t do anything. Otherwise follow these instructions:

  1. If $ n$ is even, divide by $ 2$.
  2. If $ n$ is odd, multiply by $ 3$ and add $ 1$.

Do this to the result, and keep doing this until the result is 1. For instance, choose $ n=6$. We keep applying the steps above to get the numbers 6,3,10,5,16,8,4,2,1. The 3n+1 conjecture is that for any positive integer $ n$, this sequence always gets to 1.
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Posted by Jason Polak on 30. December 2012 · Write a comment · Categories: books, number-theory · Tags: , ,

Posting has slowed a little bit this month because of holidays, but in the last couple weeks during my visit home I decided to refresh some basic knowledge of valuation theory by going through thoroughly the book “Introduction to p-adic Numbers and Valuation Theory” by George Bachman. Naturally, I wrote this quick review.

Bachman’s book is designed to be a leisurely introduction to valuation theory and p-adic numbers. It has only 152 pages and naturally cannot be comprehensive. It is, rather, an enjoyable read that does not require much advanced knowledge, though some experience with metric spaces is certainly required to fully appreciate the later chapters on the extension of valuations.
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Posted by Jason Polak on 09. October 2012 · Write a comment · Categories: number-theory · Tags: , ,

(Or would “The Number 1, Part e” be more interesting?!)

Let’s talk about the number $ e$, my favourite number. Of course, to talk about it we need a definition, so we define $ e$ as

$ e = 1/0! + 1/1! + 1/2! + 1/3! + \cdots.$

Exercise: check that this converges! In Part 2, we shall see Hermite’s argument that $ e$ is transcendental. In order to warm up, let us prove that $ e$ is not an integer first, and then let us prove that $ e$ irrational.
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