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My numbers, my friends : popular lectures on number theory

Author: Paulo Ribenboim
Publisher: New York : Springer, ©2000.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

This new collection includes essays on Fibonacci numbers, prime numbers, Bernoulli numbers, and historical presentations of the main problems pertaining to elementary number theory, such as Kummers  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Paulo Ribenboim
ISBN: 0387989110 9780387989112
OCLC Number: 42061066
Description: ix, 375 pages : illustrations ; 24 cm
Contents: The fibonacci numbers and the Artic Ocean --
Basic definitions --
Lucas sequences --
Special Lucas sequences --
Generalizations --
Basic properties --
Binet's formulas --
Degenerate Lucas sequences --
Growth and numerical calculations --
Algebraic relations --
Divisibility properties --
Prime divisors of Lucas sequences --
Prime divisors of Lucas sequences --
The sets P(U), P(V), and the rank of appearance --
Primitive factors of Lucas sequences --
Primes in Lucas sequences --
Powers and powerful numbers in Lucas sequences --
General teorems for powers --
Explicit determination in special sequences --
Uniform explicit determination of multiples, squares, and square-classes for certain families of Lucas sequences --
Powerful numbers in Lucas sequences --
Representation of real numbers by means of Fibonacci numbers --
Prime number records --
Selling primes --
Euler's famous prime generating polynomial --
Quadratic extensions --
Rings of integers --
Discriminant --
Decomposition of primes --
Properties of the norm --
Units --
The class number --
Calculation of the class number --
Determination of all quadratic fields with class number 1 --
The main theorem --
Gauss and the Class Number Problem --
Introduction --
Highlights of Gauss' life --
Brief historical background --
Binary quadratic forms --
The fundamental problems --
Equivalence of forms --
Conditional solution of the fundamental problems --
Proper equivalence classes of definite forms --
Another Numerical example --
Proper equivalence classes of indefinite forms --
Another numerical example --
The automorph of a primitive form --
Composition of proper equivalence classes of primitive forms --
The theory of genera --
The group of proper equivalence classes of primitive forms --
The theory of genera --
The group of proper equivalence classes of primitive forms --
Calculations and conjectures --
The aftermath of Gauss (or the "math" after Gauss) --
Forms versus ideals in quadratic fields --
Dirichlet's class number formula --
Solution of the class number problem for definite forms --
The class number problem for indefinite forms --
More questions and conjectures --
Many topics have not been discussed --
Consecutive powers --
Introduction --
History --
Special cases --
Divisibility properties --
Estimates --
Final comments and applications --
1093 --
Powerless facing powers --
Powerful numbers --
Distribution of powerful numbers --
Additive problems --
Difference problems --
Powers --
Pythagorean triples and Fermat's problem --
Variants of Fermat's problem --
The conjecture of Euler --
Powers as values of polynominals --
Exponential congruences --
The Wieferich congruence --
Primitive factors --
Dream mathematics --
The statements --
Statements --
Binomials and Wieferich congruences --
Erdos conjecture and Wieferich congruence --
The dream in the dream --
Kinds of numbers --
How numbers are given --
Brief historical survey --
Continued fractions --
Generalities --
Periodic continued fractions --
Approximation by rational numbers --
The order of approximation --
The Markoff numbers --
Measures of irrationality --
Order of approximation of irrational algebraic numbers --
Irrationality of special numbers --
Transcendental numbers --
Liouville numbers --
Approximation by rational numbers: sharper theorems --
Hermite, Lindemann, and Weierstrass --
A result of Siegel on exponentials --
Hilbert's 7th problem --
The work of Baker --
The conjecture of Schanuel --
Transcendence measure and the classification of Mahler.
Responsibility: Paulo Ribenboim.
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