Bertrandova domneva

magistrsko delo

Urša Blažič (Author), Aleš Vavpetič (Mentor)

Abstract

V magistrskem delu obravnavamo Bertrandovo domnevo, ki pravi, da za vsako naravno število $n$ obstaja vsaj eno praštevilo $p$, za katerega velja $n < p \leq 2n$. Podrobneje predstavimo nekaj najbolj znanih dokazov Bertrandove domneve - Erdősev, Ramanujanov in poenostavljen Ramanujanov dokaz. Erdősev dokaz temelji na oceni binomskega koeficienta, Ramanujanov dokaz pa izhaja iz prvega dokaza Bertrandove domneve, Čebiševega dokaza iz leta 1852. Poenostavljen Ramanujanov dokaz sta zapisala avtorja Meher in Ram Murty, ki sta domnevo dokazovala na enak način kot Ramanujan, le da sta se izognila uporabi Stirlingove formule. Opišemo tudi nekaj modifikacij Bertrandove domneve in njihovih uporab, med drugim Ramanujanova praštevila, ki jih Ramanujan uvede na koncu dokaza Bertrandove domneve.

Keywords

matematika;Bertrandova domneva;praštevila;Ramanujanova praštevila;praštevilski izrek;

Data

Language:	Slovenian
Year of publishing:	2022
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[U. Blažič]
UDC:	511
COBISS:	124214019
Views:	453
Downloads:	71
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Bertrand's postulate
Secondary abstract:	In the master's thesis, we study Bertrand's postulate, which states that for any given natural number $n$, there exists at least one prime number $p$ such that $n < p \leq 2n$. We present in more detail some of the most famous proofs of Bertrand's postulate - Erdős's, Ramanujan's and simplified Ramanujan's proof. Erdős's proof is based on estimation of the binomial coefficient. Ramanujan's proof derives from the first proof of Bertrand's postulate, Chebyshev's proof from 1852. Ramanujan's simplified proof was written by Meher and Ram Murty, who proved the postulate in the same way as Ramanujan, except they avoided using Stirling's formula in their proof. We also describe modifications of Bertrand's postulate and their applications, including Ramanujan's primes that Ramanujan introduces at the end of the proof of Bertrand's postulate.
Secondary keywords:	mathematics;Bertrand postulate;prime numbers;Ramanujan primes;prime number theorem;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Pedagoška matematika
Pages:	IX, 58 str.
ID:	16653076