# Fermatovi brojevi

Lepan, Ivana (2015) Fermatovi brojevi. Diploma thesis, Faculty of Science > Department of Mathematics.

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## Abstract

Numbers of the form $f_n = 2^{2n} + 1, n = 0, 1, 2, \dots$ are called Fermat numbers. Prime Fermat numbers are called Fermat primes. Fermat conjectured that all numbers of that form are prime. However, Euler disproved the Fermat's conjecture when he showed that $f_5$ is composite. So, the question is: Are there infinitely many prime numbers of the form $2^{2n} + 1, n \in \mathbb{N}$? The conjecture states that the set of prime Fermat numbers is finite. However, that is an open problem to which we haven't found an answer. Furthermore, the study of Fermat numbers has opened many other problems which are yet to be solved. In this paper we have named some of the well known ones. The aim of this work is not to put the emphasis on open problems, but on the results found by many well known mathematicians while studying Fermat numbers. So, in this work we consider some basic properties of Fermat numbers, give some primality tests for Fermat numbers and describe factors of composite Fermat numbers.

Item Type: Thesis (Diploma thesis) Franušić, Zrinka 2015 41 NATURAL SCIENCES > Mathematics Faculty of Science > Department of Mathematics Iva Prah 21 Oct 2015 08:15 18 Apr 2016 09:31 http://digre.pmf.unizg.hr/id/eprint/4154

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