58 research outputs found
Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
In 1862 Wolstenholme proved that for any prime the numerator of the
fraction written in reduced form is divisible by , and the numerator of
the fraction
written in reduced form is divisible by . The first of the above
congruences, the so called {\it Wolstenholme's theorem}, is a fundamental
congruence in combinatorial number theory. In this article, consisting of 11
sections, we provide a historical survey of Wolstenholme's type congruences and
related problems. Namely, we present and compare several generalizations and
extensions of Wolstenholme's theorem obtained in the last hundred and fifty
years. In particular, we present more than 70 variations and generalizations of
this theorem including congruences for Wolstenholme primes. These congruences
are discussed here by 33 remarks.
The Bibliography of this article contains 106 references consisting of 13
textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of
Integer Sequences. In this article, some results of these references are cited
as generalizations of certain Wolstenholme's type congruences, but without the
expositions of related congruences. The total number of citations given here is
189.Comment: 31 pages. We provide a historical survey of Wolstenholme's type
congruences (1862-2012) including more than 70 related results and 106
references. This is in fact version 2 of the paper extended with congruences
(12) and (13
Constellations and multicontinued fractions: application to Eulerian triangulations
We consider the problem of enumerating planar constellations with two points
at a prescribed distance. Our approach relies on a combinatorial correspondence
between this family of constellations and the simpler family of rooted
constellations, which we may formulate algebraically in terms of multicontinued
fractions and generalized Hankel determinants. As an application, we provide a
combinatorial derivation of the generating function of Eulerian triangulations
with two points at a prescribed distance.Comment: 12 pages, 4 figure
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
Diophantine equations with arithmetic functions and binary recurrences sequences
A thesis submitted to the Faculty of Science, University of the
Witwatersrand and to the University Cheikh Anta Diop of Dakar(UCAD)
in fulfillment of the requirements for a Dual-degree for Doctor in
Philosophy in Mathematics. November 6th, 2017.This thesis is about the study of Diophantine equations involving binary recurrent
sequences with arithmetic functions. Various Diophantine problems are investigated
and new results are found out of this study. Firstly, we study several
questions concerning the intersection between two classes of non-degenerate binary
recurrence sequences and provide, whenever possible, effective bounds on
the largest member of this intersection. Our main study concerns Diophantine
equations of the form '(jaunj) = jbvmj; where ' is the Euler totient function,
fungn 0 and fvmgm 0 are two non-degenerate binary recurrence sequences and
a; b some positive integers. More precisely, we study problems involving members
of the recurrent sequences being rep-digits, Lehmer numbers, whose Euler’s
function remain in the same sequence. We prove that there is no Lehmer number
neither in the Lucas sequence fLngn 0 nor in the Pell sequence fPngn 0. The
main tools used in this thesis are lower bounds for linear forms in logarithms
of algebraic numbers, the so-called Baker-Davenport reduction method, continued
fractions, elementary estimates from the theory of prime numbers and sieve
methods.LG201
- …