## Archive for January, 2022

### Gaussian Primes

We are all familiar with splitting natural numbers into prime components. This decomposition is unique, except for the order of the factors. We can apply the idea of prime components to many more general sets of numbers.

The Gaussian integers are all the complex numbers with integer real and imaginary parts, that is, all numbers in the set

$\displaystyle \mathbb{Z}[i] \equiv \{ m + i n : m, n \in \mathbb{Z} \} \,.$

The set ${\mathbb{Z}[i]}$ forms a two-dimensional lattice in the complex plane. For any element ${g \in \mathbb{Z}[i]}$ we consider the four numbers ${\{g, -g, ig, -ig \}}$ as associates. The associates of ${1}$ are known as units: ${\{1, -1, i, -i \}}$.

### Letters to a German Princess: Euler’s Blockbuster Lives On

The great Swiss mathematician Leonhard Euler produced profound and abundant mathematical works. Publication of his Opera Omnia began in 1911 and, with close to 100 volumes in print, it is nearing completion. Although he published several successful mathematical textbooks, the book that attracted the widest readership was not a mathematical work, but a collection of letters  [TM227 or search for “thatsmaths” at irishtimes.com].

For several years, starting in 1760, Euler wrote a series of letters to Friederike Charlotte, Princess of Brandenburg-Schwedt, a niece of Frederick the Great of Prussia. The collection of 234 letters was first published in French, the language of the nobility, as Lettres à une Princesse d’Allemagne. This remarkably successful popularisation of science appeared in many editions, in several languages, and was widely read. Subtitled “On various subjects in physics and philosophy”, the first two of three volumes were published in 1768 by the Imperial Academy of Sciences in St. Petersburg, with the support of the empress, Catherine II.

### Euler’s Journey to Saint Petersburg

It all began with an invitation to Leonhard Euler to accept a chair of mathematics at the new Imperial Academy of Science in the city founded by Peter the Great. Euler’s journey from Basel to Saint Petersburg was a highly influential factor for the development of the mathematical sciences. The journey is described in detail in a full-length biography of Euler by Ronald Calinger (2016). The account below is heavily dependent on Calinger’s book.

Drawing based on a map of Europe from about 1740 (from Calinger, 2016, pg. 39). Euler’s route from Basel to Saint Petersburg is marked by the heavy dashed line.

### Some Characteristics of the Mathematical Psyche

What are mathematicians really like? What are the characteristics or traits of personality typical amongst them?  Mathematicians are rarely the heroes of novels, so we have little to learn from literature. A few films have featured mathematicians, but most give little insight into the personalities of their subjects [TM226 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Some Characteristics of the Mathematical Psyche’