Image for post
Image for post

And the Geometry of Numbers

This is part two of a series about Leonard Euler (1707–1783). You can read the first part here.

Before we begin our quest towards understanding the beautiful fact called Euler’s Identity, let’s warm up with an amazing piece of history.

Numbers in Love

About 500 b.c. the Greeks regarded some numbers as more important than others. In particular, they knew of two numbers with a remarkable property. The two numbers are 220 and 284.

Before explaining why these numbers are so interesting, we need to know what a proper divisor is. Well, it is very simple. A proper divisor of say, n, is…


The nature of symmetry and the symmetry of nature

Image for post
Image for post
Image by geralt on Pixabay

Group theory is the language of many of the mathematical disciplines. An indispensable tool in understanding the underlying nature of nature.

A theory that holds the secrets of the fundamental particles and forces of the Universe itself!

We use it to understand shapes in higher dimensions, the proof of insolvability of higher degree polynomials, and the structure of number systems, but the use of this beautiful theory doesn’t stop there.

Group theory is the study of symmetry and as you will soon see, symmetry is everywhere in mathematics and in nature.

My own encounter with groups started when I studied…


Generalizing Leibniz' formula for π, the alternating harmonic series, and the Basel problem

Image for post
Image for post
Image by author

A couple of years ago I was thinking about how to generalize many of the infinite series that we know so well. It was clear to me that it was possible to generalize them since some of my studies into the field of Fourier series had led me to some of these generalizations in the past. But it was clear that I needed another way - a more controlled way to generate such formulas.

Not long ago, I found a method and I couldn’t sleep that night because I was calculating like crazy in my head. …


An ancient riddle

Image for post
Image for post
Photo by Clem Onojeghuo on Unsplash

In mathematics, especially when it comes to number theory, there are many basic things that we don’t know.

Simple questions that we can’t answer.

This can be realized by considering a few of the famous unanswered questions that we mathematicians would like to know the answer to.

A few of them are

  • Are there infinitely many primes p such that p+2 is also prime?
  • Are there infinitely many Mersenne primes?
  • Can every even number greater than 2 be written as a sum of two primes?
  • Are there infinitely many amicable numbers?
  • Is the Collatz conjecture true?

These questions seem deceptively…


Use Neo4j to calculate a safety score — how safe is your code?

Image for post
Image for post
Image by author

As a mathematician and working data scientist, I am fascinated by programming languages, machine learning, data, and of course, mathematics.

These technologies, arts and tools, are of course of huge importance to society and they are transforming our lives as you read this article, but another emerging technology is growing. And it is growing fast!

It is a technology based on a mathematical field that I studied at university and which was first discovered (or invented..? let’s safe that talk for another time, shall we?) …


Capture the Essence of an Equation and Solve it Visually by Using Mathematical Graphs

Image for post
Image for post
“Normal” Mathematical Equations

Before we get started on this rather visual investigation into the nature of equations, we need to recall that an equation is a relation and that relation can be written down in more than one way.

Throughout this article, it is important that you let go and forget about the old conventions that you learned long ago because this is a completely new (at least for me) way of thinking.

Introduction

At school, many of us learned that an equation is like a old scale in balance.

You need to keep the balance, so if you add a number on one…


Number Mysteries and the Hunt for Large Primes

Image for post
Image for post
Marin Mersenne

In this short introductory article, I will take you on a little journey into the exotic world of prime numbers.

It all started in France with the birth of a boy, that was to become one of the greatest scientists, philosophers, and mathematicians of his time.

Marin Mersenne was born on 8 September 1588. He was a priest, scientist, and mathematician studying everything of interest from vibrating strings on musical instruments to different fields of mathematics.

Mersenne was in a sense a center of science at his time corresponding with Galileo Galilei, René Descartes, Étienne Pascal, Pierre de Fermat, among…


An Equivalent Formulation of an Unsolved Mystery

Image for post
Image for post
Even numbers as sums of two primes

The Goldbach conjecture is one of the most captivating unsolved mysteries in mathematics. In this article, I will take you on a journey through time and mathematics.

I will present some other ways of looking at the conjecture than the original definition. Both visually and algebraically. We will prove an equivalence and investigate it a bit.

It’s All About the Primes

The Goldbach conjecture is about prime numbers so I believe that a word about them is in order.

Before jumping headfirst into one of the oldest and most “feared” questions in mathematics, let’s try to understand why we should care about questions involving prime…


You Don’t Need to Remember the Formula

Image for post
Image for post
Photo by Willian Justen de Vasconcellos on Unsplash

Suppose you don’t remember the quadratic formula and you have to solve a quadratic equation of the form


Image for post
Image for post

A Probabilistic Approach

The Euler totient function plays an important role in the field of number theory so I’ve decided to give a brief introduction to it in this article with some nice applications.

Coprime Integers

We say that an integer m divides n (or is a divisor of n) if n/m is a whole number and we write m|n. Furthermore, we say that two whole numbers m and n are relatively prime (or coprime) if the greatest common divisor is 1. In this case, we write (n,m) = 1. In the literature, you will sometimes see the notation gcd(n,m) instead of (n,m).

For example…

Kasper Müller

Mathematician and Data Scientist interested in the mysteries of the Universe, fascinated by the human mind, music and things that I don’t understand.

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store