Michal Křížek - From Great Discoveries in Number Theory to Applications

For us, mathematical theorems and their proofs

are like gold nuggets to a prospector.

T he authors

We encounter integer numbers daily, and they are literally everywhere around us. It is not possible to avoid them, ignore them, or to be indifferent to them. So let us take together a journey through the world of integers, to get acquainted with their fascinating and sometimes magic properties. We will discover some surprising connections between number theory and geometry (see, e.g., Chaps. ). We shall see which laws are followed by integers. We will also show that number theory has many practical applications without which we could not imagine the modern technical world. It has a big influence on everything we do.

This treatise on integer numbers is based on our more than 70 works on elementary and algebraic number theory that we published between the years 2001 and 2021 mostly in prestigious international journals such as Journal of Number Theory, Integers, The Fibonacci Quarterly, Discrete Mathematics, Journal of Integer Sequences, Proceedings of the American Mathematical Society, and Czechoslovak Mathematical Journal (see, e.g., dml.cz). Most of our results were reported at many international conferences on number theory and also the regular Friday seminar Current Problems in Numerical Analysis, which takes place at the Institute of Mathematics of the Czech Academy of Sciences in Prague [426].

The book is intended for a general mathematical audienceespecially for those who can appreciate the beauty of both abstract and applied mathematics. We only assume that the reader is familiar with the basic rules of arithmetic and has no problem with adjustments of algebraic formulas. Only very rarely it is necessary to understand some relationships from linear algebra or calculus. Most chapters can be read independently from one another. Some parts are quite simple, others more complicated. If some part is too difficult, there is no problem in skipping it.

At the end of the book, there are several tables and a fairly extensive bibliography to attract attention to some important works in number theory. For inspiration, there are also several links to websites, although we are well aware that they are not subjected to any review and change quite frequently. Newly defined terms are highlighted in italics in the text for the convenience of the reader. They can also be found in the Index.

In order to read the individual chapters, it is not necessary that the reader understands all theorems. There are 230 of them. Mathematicians formulate their ideas in the form of mathematical theorems that contain only what is relevant in the problem in question. We provide proofs of most statements so that one can verify their validity. For more complicated proofs, we only give a reference to the corresponding literature. The most beautiful feature of number theory is that the main ideas of proofs of every statement usually differ from each other. Formulations of mathematical theorems presented in this book often take only one or two lines, which makes it relatively easy to understand what a particular theorem says.

Mathematical theorems are valid forever. They are independent of position and time. Parliament does not decide about their validity by voting, nor the religious or political system in some country, nor does it depend on cultural customs. For example, the famous Pythagorean Theorem is valid on Earth as well as on the distant Andromeda galaxy M31, and it will also be valid after millions of years. Definitions of mathematical terms do not allow a double meaning. Also absolutely accurate formulations of mathematical problems do not allow more interpretations. The vague expressions we witness in daily life lead to a number of misunderstandings. Only a small percentage of our population is able to express their ideas accurately and perceive the beauty of mathematics. This was aptly stated by the well-known Hungarian mathematician Cornelius Lanczos (18931974) as follows:

Most of the arts, as painting, sculpture, and music, have emotional appeal to the general public. This is because these arts can be experienced by one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians, and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose.

There are many books on number theory. Let us mention, e.g., [6, 9, 28, 56, 82, 85, 91, 127, 132, 137, 138, 176, 235, 284, 291, 321324, 327, 333, 344, 347, 350, 395, 413, 424]. However, our book contains some nonstandard topics. For instance, we will see how triangular numbers are related to the bell-work machinery of the Prague Astronomical Clock, what kind of mathematics is hidden in the traditional Chinese calendar, how the Fundamental Theorem of Arithmetic was used to design a message to extraterrestrial civilizations, how number theory is related to chaos, fractals, and graph theory. We will construct a