Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.Comment: 25 page

Primality Tests on Commutator Curves

Das Thema dieser Dissertation sind effiziente Primzahltests. Zunächst wird die Kommutatorkurve eingeführt, die durch einen skalaren Parameter in der zweidimensionalen speziellen linearen Gruppe bestimmt wird. Nach Erforschung der Grundlagen dieser Kurve wird sie in verschiedene Pseudoprimzahltests (z.B. Fermat-Test, Solovay-Strassen-Test) eingebunden. Als wichtigster Pseudoprimzahltest ist dabei der Kommutatorkurventest zu nennen. Es wird bewiesen, dass dieser Test nach einer festen Anzahl von Probedivisionen (alle Primzahlen kleiner 80) das Ergebnis 'wahr' für eine zusammengesetzte Zahl mit einer Wahrscheinlichkeit ausgibt, die kleiner als 1/16 ist. Darüberhinaus wird bewiesen, dass der Miller-Primzahltest unter der Annahme der Korrektheit der Erweiterten Riemannschen Hypothese zur Überprüfung einer Zahl n nur noch für alle Primzahlbasen kleiner als 3/2*ln(n)^2 durchgeführt werden muss. Im Beweis des Primzahltests von G. L. Miller konnte dabei die Notwendigkeit der Erweiterten Riemannschen Hypothese auf nur noch ein Schlüssellemma eingegrenzt werden.This thesis is about efficient primality tests. First, the commutator curve which is described by one scalar parameter in the two-dimensional special linear group will be introduced. After fundamental research of of this curve, it will be included into different compositeness tests (e.g. Fermat's test, Solovay-Strassen test). The most important commutator test is the Commutator Curve Test. Besides, it will be proved that this test after a fixed number of trial divisions (all prime numbers up to 80) returns the result 'true' for a composite number with a probability less than 1/16. Moreover, it will be shown that Miller's test to check a number n only has to be carried out for all prime bases less than 3/2*ln(n)^2. This happens under the assumption that the Extended Riemann Hypothesis is true. The necessity of the Extended Riemann Hypothesis to prove the primality test of G. L. Miller can be reduced to a single key lemma

On Generating Prime Numbers Efficiently

The prime numbers can be considered as the building blocks of natural numbers, having innumerable applications in number theory and cryptography. There exist multiple different sieving algorithms for the generation of prime numbers. In this thesis, an elementary modular result is utilized to construct an analytically useful generator function and its inverse function. The functions are used to generate a (log)log-linear time complexity prime sieving algorithm which is further optimized to be of linear time complexity. The constructed algorithms and their operation are studied and the linear implementations in JS, Python and C++ are compared to other prime sieves.Alkulukuja voidaan pitää luonnollisten lukujen rakennuspalikoina joilla on lukemattomia sovelluksia lukuteoriassa ja kryptografiassa. Alkulukujen luomiseen on olemassa useita erilaisia seulonta-algoritmeja. Tässä opinnäytetyössä käytetään modulaarista perustulosta analyyttisesti hyödyllisten kehitysfunktion ja sen käänteisfunktion luomiseen. Funktioiden avulla luodaan aikakompleksisuudeltaan (log)log-lineaarinen alkulukuseula, joka optimoidaan lineaariseksi. Rakennettuja algoritmeja ja niiden toimintaa tarkastellaan ja lineaarista implementaatiota JS, Python ja C++ ohjelmointikielillä verrataan toisiin alkulukuseuloihin

Mersenne numbers

These notes have been issued on a small scale in 1983 and 1987 and on request at other times. This issue follows two items of news. First, WaIter Colquitt and Luther Welsh found the 'missed' Mersenne prime M110503 and advanced the frontier of complete Mp-testing to 139,267. In so doing, they terminated Slowinski's significant string of four consecutive Mersenne primes. Secondly, a team of five established a non-Mersenne number as the largest known prime. This result terminated the 1952-89 reign of Mersenne primes. All the original Mersenne numbers with p < 258 were factorised some time ago. The Sandia Laboratories team of Davis, Holdridge & Simmons with some little assistance from a CRAY machine cracked M211 in 1983 and M251 in 1984. They contributed their results to the 'Cunningham Project', care of Sam Wagstaff. That project is now moving apace thanks to developments in technology, factorisation and primality testing. New levels of computer power and new computer architectures motivated by the open-ended promise of parallelism are now available. Once again, the suppliers may be offering free buildings with the computer. However, the Sandia '84 CRAY-l implementation of the quadratic-sieve method is now outpowered by the number-field sieve technique. This is deployed on either purpose-built hardware or large syndicates, even distributed world-wide, of collaborating standard processors. New factorisation techniques of both special and general applicability have been defined and deployed. The elliptic-curve method finds large factors with helpful properties while the number-field sieve approach is breaking down composites with over one hundred digits. The material is updated on an occasional basis to follow the latest developments in primality-testing large Mp and factorising smaller Mp; all dates derive from the published literature or referenced private communications. Minor corrections, additions and changes merely advance the issue number after the decimal point. The reader is invited to report any errors and omissions that have escaped the proof-reading, to answer the unresolved questions noted and to suggest additional material associated with this subject

Carmichael numbers

RESUMEN: Este trabajo pretende ser una guía para que el lector se introduzca en el estudio de los números de Carmichael. Se comienza introduciendo varias nociones que conducen, de forma natural, a la definición de esta familia de números. Posteriormente, se estudian algunas de sus propiedades más elementales y algunos procedimientos para generarlos. El texto termina discutiendo resultados concernientes a la distribución de los números de Carmichael. A lo largo del trabajo, se discuten algunas de las conjeturas y problemas abiertos relacionadas con estos números, así como posibles líneas de investigación. Como un objetivo secundario, este trabajo busca que el lector interesado pueda familiarizarse con técnicas y resultados al uso en el estudio de la teoría de números.ABSTRACT: This work aims to be a guide for the reader to introduce to the study of Carmichael numbers. Firstly, we introduce several concepts that lead to a natural definition of this family of numbers. Later on, we study some of the most relevant properties regarding these numbers, as well as some procedures to generate them. The text ends with the discussion of results concerning the distribution of Carmichael numbers. Throughout this document, some of the conjectures and open problems related to these numbers are brought up. As a secondary objective, this work allows the interested reader to become familiar with some habitual techniques and results to be used in number theory.Grado en Matemática

Asymptotics For Primitive Roots Producing Polynomials And Primitive Points On Elliptic Curves

Let $x \geq 1$ be a large number, let $f(n) \in \mathbb{Z}[x]$ be a prime producing polynomial of degree $\deg(f)=m$, and let $u\neq \pm 1,v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes $p=f(n) \leq x$ with a fixed primitive root $u$ is derived in this note. This asymptotic result has the form \pi_f(x)=\# \{ p=f(n)\leq x:\ord_p(u)=p-1 \}=\left (c(u,f)+ O\left (1/\log x )\right ) \right )x^{1/m}/\log x, where $c(u,f)$ is a constant depending on the polynomial and the fixed integer. Furthermore, new results for the asymptotic order of elliptic primes with respect to fixed elliptic curves $E:f(X,Y)=0$ and its groups of $\mathbb{F}_p$-rational points $E(\mathbb{F}_p)$, and primitive points are proved in the last chapters

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Números de Carmichael

Exposición de los resultados más comunes sobre pseudoprimos y números de Carmichael. Estos números, si bien no son primos, tienen en muchos aspectos un comportamiento similar a los números primos. Se presentan diversos métodos para generarlos. Se estudia su estructura. Se presentan teoremas y conjeturas sobre su existencia.Grado en Matemática