 | The course contains 9 parts. The first part deals with divisibility as one of the basic concepts in number theory. In this section, terms such as the “divides” relationship, the greatest common divisor or the least common multiple are explained. In the second part, we have included basic knowledge about prime numbers and we explain their position among numbers. The part contains several key terms pertaining to prime numbers, such as the fundamental theorem of arithmetic or canonical decomposition of an integer. The content of the third part contains solution to linear Diophantine equations in two unknowns and Fermat’s Last Theorem. The part also explains the principle of division with a remainder, the so called Euclidean algorithm. part four explains the principle of inclusion and exclusion, using various applications, which is applicable e.g. when deriving Euler’s function. part five is dedicated to arithmetic functions and perfect numbers. Using arithmetic functions, we can determine to which of three sets (abundant, redundant, perfect) a given number belongs. Using arithmetic functions, we can determine and search for various other properties of numbers and they also have their own special position in number theory. part six consists of Euler’s theorem and Fermat’s Little Theorem, which are of great use in number theory. Euler’s theorem is currently widely used in computer science and cryptography, therefore good knowledge of the theorem and its use in specific mathematical applications is important. In the next seventh part, we deal with several special applications pertaining to numeral systems. In this section, we discuss some important basic knowledge from the field of numeral system theory pertaining to notation of numbers in systems or to criteria for divisibility which hold in general for systems with any number as their base. The eighth part consists of residue classes and linear congruences. We focused mainly on the basic properties of congruences and on solving congruences and their systems. In this part, we explained the necessary and sufficient condition for solvability of a system, the so called Chinese remainder theorem. Fibonacci numbers play an important role in mathematics and appear in many places when solving practical problems even nowadays, which is why we included them in the course under part nine. |
