Teorija brojeva kroz povijest

Gmajnić, Ana (2015) Teorija brojeva kroz povijest. Diploma thesis, Faculty of Science > Department of Mathematics.

[img]
Preview
PDF
Language: Croatian

Download (352kB) | Preview

Abstract

Number theory is one of the oldest branches of mathematics, and there is evidence that it has been around for some 4500 years. Early history starts with Plimpton 322 tablet on which we can find the first traces of number theory. After that, many theorems can be found in Pythagoras work (500 BC). Pythagoras was trying to find positive integers $x, y, z$ for which $x^2 + y^2 = z^2$ is valid. In Euclid’s Elements (300 BC) we can find many theorems that are related to divisibility and prime numbers. Also, we can find the Euclidean algorithm which is used, even today, for finding greatest common divisors. Euclid also mentions the fundamental theorem of arithmetic, which is proven by Gauss in the 19th century. After Euclid, number theory enters a dormant period which lasts until the 16th century, when it once again peaks interest amongst mathematicians. Noteworthy mathematicians in this period are Marin Mersenne and Pierre de Fermat who studied the properties of prime numbers. Fermat was responsible for two great theorems of that time, Fermat’s little theorem and Fermat’s last theorem. Fermat’s little theorem was proven by Leonhard Euler in 1736, and Fermat’s last theorem was proven by Andrew Wiles in 1994. Leonhard Euler is considered to be one of the most prolific mathematicians in history, and is responsible for studying number theory trough the methods of mathematical analysis. Euler’s contemporary Christian Goldbach came forward with a conjecture that stated: Any number greater than two can be written as the sum of two prime numbers. This conjecture has not yet been proven. Important mathematicians of the 18th century were Joseph Louis Lagrange and Johann Friedrich Gauss. Lagrange is responsible for proofs of many theorems, including Wilson’s theorem and the theorem of four squares. Gauss is responsible for establishing the theory of congruences and for the proof of the Law of Quadratic Reciprocity. Also, he expanded the set of integers to the set of so-called Gaussian integers. Finally, in the 19th century we have two important mathematicians, Legendre and Dirichlet. Legendre, among other things, is known for the prime number theorem, while Dirichlet is known for Dirichlet theorem on primes in arithmetic progression.

Item Type: Thesis (Diploma thesis)
Supervisor: Najman, Filip
Date: 2015
Number of Pages: 52
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 09 Oct 2015 09:38
Last Modified: 09 Oct 2015 09:38
URI: http://digre.pmf.unizg.hr/id/eprint/4265

Actions (login required)

View Item View Item