The Prime Bet

 Posted on May 31, 2017  |  9 minutes  |  1723 words  |  Markus Shepherd

Let’s say you sit in a pub, minding your own business, when all of a sudden a stranger walks up to you and offers you a bet: We’ll choose two positive integers at random. If they have any divisor in common (other than \(1\)) I’ll pay you a dollar, else you’ll pay me a dollar. Are you in? Apart from the question what kind of establishments you frequent, you should be wondering: is this a good bet for you? [Read More]
Apéry  Basel problem  Bet  constant  Coprime  Divisor  Euler  gcd  pi  Prime factor  Probability  Pub  Python  Random  Riemann  Wiles  Zeta function  Zhang 

Numbers of the World

 Posted on October 20, 2016  |  9 minutes  |  1841 words  |  Markus Shepherd

Recently Matt Parker uploaded a video to his YouTube channel where he discussed numbers and the words used to represent them in different languages, more precisely the length of these words: The basic idea is the following: one has 3 letters, two has 3 letters, three has 5 letters, four has 4 letters, five has 4 letters, six has 3 letters, seven has 5 letters, eight has 5 letters, nine has 4 letters, ten has 3 letters, and so on… This can be seen as a function [Read More]
binary  chain  connected component  cycle  Cypher  Danish  English  Esperanto  fixed point  French  German  GitHub  Graph  Indonesian  languages  Latvian  letters  literumi  Lithuanian  Matt Parker  Neo4j  nombroj  Norwegian  numbers  path  Polish  Portuguese  Python  recreational maths  Roman numerals  Russian  Spanish  spelling  words  world  YouTube 

Primes from a Different World

 Posted on August 5, 2016  |  4 minutes  |  687 words  |  Markus Shepherd

Every textbook on number theory will begin with a treatise on prime numbers; every treatise on prime numbers will begin by emphasising their importance as building blocks or atoms of our number system: every integer can be expressed as a product of prime numbers in one way and one way only. Six is two times three and there is no other way to decompose it.1 Euclid proved this over two thousand years ago and it is so fundamental (hence the name fundamental theorem of arithmetic) to our thinking about numbers that we take it for granted. [Read More]
algebraic number theory  factorisation  Fermat’s Last Theorem  Fundamental Theorem  Gauss  Gaussian integer  integers  Kummer  Lamé  Number theory  prime  unique prime factorisation  Wiles 

Visualising the Riemann Hypothesis

 Posted on April 10, 2016  |  3 minutes  |  429 words  |  Markus Shepherd

Update 2024-10-20: Watch the video now in 4K! One stubborn source of frustration when working with complex numbers is the fact that visualisation becomes tedious, if not impossible. Complex numbers have 2 “real” dimensions in themselves, which give rise to the complex plane. That’s all good and fair. But if you talk about a function with complex domain and codomain, you already deal with a 4-dimensional graph. Unfortunately, my mind can only handle 3 dimensions (on a good day). [Read More]
Complex numbers  Dimensions  Graph  Sage  SageMath  Video  Visualisation  Zeta function 

A Toy Keyless Encryption Protocol

 Posted on October 2, 2015  |  7 minutes  |  1486 words  |  Markus Shepherd

Cryptography is a natural application of number theory and so I’d like to write down a few thoughts about it in this blog. (The fact that there are real world applications to number theory deserves some appreciation in itself, but it would throw us too much off track here.) One particularly nice feature of cryptography is the ability to explain its inner workings with real world analogies about security. For instance, one way two parties (who we, by convention, call Alice and Bob) could hide their secrete communication is if Alice writes a letter, puts it in a box, and locks it with a padlock for which both she and Bob have a key, but no one else. [Read More]
Alice  Bob  Elliptic Curves  Eve  Groups  Information Theory  Lattice  Number theory  OTP  Primes  XOR 

Prime Generating Sequences

 Posted on September 23, 2015  |  6 minutes  |  1166 words  |  Markus Shepherd

A couple of months ago (really, a long, long time ago) I found an interesting question on Mathematics Stack Exchange (another site to effectively waste away hours of your life). It reminded me of my Bachelor’s thesis (which I wrote a really, really long time ago) about the sequence \[ g_n=\mathrm{gcd}(n,a_{n-1})=(n,a_{n-1}) \quad\text{for}\quad n>1, \] where \(a_1=7\) and \[ a_n=a_{n-1}+g_n. \] Here, \(\mathrm{gcd}(a,b)=(a,b)\) stands for the greatest common divisor,1 i.e., the largest integer \(d\) that divides both \(a\) and \(b\). [Read More]
Mathematics Stack Exchange  Paper  Primes  Rowland  Sequence  Stack Exchange  Thesis 

If One at a Time is too Difficult, Try All at Once!

 Posted on December 27, 2014  |  5 minutes  |  926 words  |  Markus Shepherd

In the past months, I spent as much time as I had on taking online courses at Coursera. One particularly interesting course, both from a mathematical and computational point of view, is Analytic Combinatorics which applies combinatorics (i.e., the art of counting) to the analysis of algorithms by finding formulae, exact or asymptotic, for their running time. It is notoriously difficult to find exact formulae for general combinatorial constructs. Typically, we want to know how many objects, e. [Read More]
Analysis  Asymptotics  Combinatorics  Coursera  Derbyshire  Dirichlet  du Sautoy  Number theory  Riemann 

Applying the Explicit Formula

 Posted on November 23, 2014  |  4 minutes  |  649 words  |  Markus Shepherd

It’s quite some time since we arrived at Riemann’s main result, the explicit formula \[ J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2, \] where \(J(x)\) is the prime power counting function introduced even earlier. It’s high time we applied this! First, let’s take a look at \(J(x)\) when calculating it exactly: You see how this jumps by one unit at prime values (\(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(17\), \(19\)), by half a unit at squares of primes (\(4\), \(9\)), by a third at cubes (\(8\)), and by a quarter at fourth powers (\(16\)), but is constant otherwise. [Read More]
Approximation  Prime counting function  Prime number theorem  Primes  Riemann  Zeros  Zeta function  Zeta zeros 

How NOT to Earn a Million Dollars

 Posted on November 2, 2014  |  4 minutes  |  771 words  |  Markus Shepherd

I recently spent some time on the formidable website Numberphile which explains mathematical ideas, some important, some recreational, in short and accessible videos. Definitely worth checking out. One of the videos that is most relevant to us explains the Riemann Hypothesis: As mentioned before, it’s not easy to explain the details and the beauty of the Riemann Hypothesis in few words, but I think the video definitely succeeds in compressing the essentials into 17 minutes. [Read More]
Bombieri  CMI  Gauss  Littlewood  Logarithm  Mertens  Millennium  Million  Numberphile  Riemann  YouTube  Zagier 

Tossing the Prime Coin

 Posted on October 12, 2014  |  7 minutes  |  1412 words  |  Markus Shepherd

One of the problems with explaining the Riemann Hypothesis is that its fascination comes from its deep connection to prime numbers, but its definition is in terms of complex analysis which requires a fair deal of undergraduate mathematics to understand – and that is before you even got started to grasp what the heck the zeta-zeros have to do with the distribution of primes. My “cocktail party explanation” of the Riemann Hypothesis would usually be something like: “The prime numbers are as equally distributed as you could wish for. [Read More]
Coin  Mertens  Möbius  Mu function  Random walk