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Anneli Lax New Mathematical Library
Invitation to Number Theory
Number theory has been instrumental in introducing many of the most distinguished mathematicains, past and present, to the charms and mysteries of mathematical research. The purpose of this simple little guide will have been achieved if it should lead some of its readers to appreciate why the properties of nubers can be so fascinating. It would be better still if it would induce you to try to find some number relations of your own; new curiosities devised by young people turn up every year. In any case, you will become familiar with some of the special mathematical concepts and methods used in number theory and will be prepared to embark upon the study of the more advanced books in its rich literature.
Contents
1. Introduction
History
Numerology
The Pythagorean problem
Figurate numbers
Magic squares
2. Primes
Primes and composite numbers
Mersenne primes
Fermat primes
The sieve of Eratosthenes
3. Divisors of numbers
Fundamental factorization theorem
Divisors
Problems concerning divisors
Perfect numbers
Amicable numbers
4. Greatest common divisor and least common multiple
Greatest common divisor
Relatively prime numbers
Euclid´s algorithm
Least common multiple
5. The Pythagorean problem
Preliminaries
Solutions of the Pythagorean equation
Problems connected with Pythagorea triangles
6. Numeration systems
Numbers for the millions
Other systems
Comparisons of numeration systems
Some problems concerning numeration systems
Computers and their numeration systems
Games with digits
7. Congruences
Definition of congruence
Some properties of congruences
The algebra of congruences
Powers of congruences
Fermat´s congruence
8. Some applications of congruences
Checks on computations
The days of the week
Tournament schedules
Prime or composite?
Solutions to selected problems
References
Index
Contents
1. Introduction
History
Numerology
The Pythagorean problem
Figurate numbers
Magic squares
2. Primes
Primes and composite numbers
Mersenne primes
Fermat primes
The sieve of Eratosthenes
3. Divisors of numbers
Fundamental factorization theorem
Divisors
Problems concerning divisors
Perfect numbers
Amicable numbers
4. Greatest common divisor and least common multiple
Greatest common divisor
Relatively prime numbers
Euclid´s algorithm
Least common multiple
5. The Pythagorean problem
Preliminaries
Solutions of the Pythagorean equation
Problems connected with Pythagorea triangles
6. Numeration systems
Numbers for the millions
Other systems
Comparisons of numeration systems
Some problems concerning numeration systems
Computers and their numeration systems
Games with digits
7. Congruences
Definition of congruence
Some properties of congruences
The algebra of congruences
Powers of congruences
Fermat´s congruence
8. Some applications of congruences
Checks on computations
The days of the week
Tournament schedules
Prime or composite?
Solutions to selected problems
References
Index
- GenresMathematics
129 pages, Paperback
First published June 1, 1967
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Displaying 1 - 2 of 2 reviews
July 14, 2017
It's a good book, but ta not quite as amazing as his graph theory book.
November 21, 2012
A charming and very readable introduction to number theory perfectly pitched for readers with intermediate high school knowledge of mathematics (a basic knowledge of divisibility and of Pythagoras´ theorem is probably sufficient to carry the reader through). The editors state that: "This book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and readable to a large audience of high school students and laymen." Measured by this goal, I believe the book is wonderfully successful and hence the five stars in my rating.
The text niftily and succintly weaves some interesting history, beguiling mathematical problems and, in general, clear proofs of key theorems and assertions together with a smattering of problems to induce the reader to work out and understand the a well chosen set of related mathematical concepts. The books is an entertaining stroll which leads up to the solutions of the Pythagorean equation (for which integer values of the three sides are there right-angled triangles?) and to Fermat´s congruence (also known as the little Fermat´s theory) but pointing out important features on the way such as magic squares, Mersenne and Fermat primes, perfect and amicable numbers, numeration systems, the algebra of congruences and its application to checks on computations, finding out the day of the week any past or future date falls and how to build round-robin tournament schedules, amongst others. Written in 1967, it is a bit dated as regards some aspects of the field such as, understandably, the largest primes calculated using computers and I´m sure that, were Professor Ore (1899-1968) alive today and updating the book, he would carefully add a little something about the applications of number theory to cryptography. However, this does not detract from the merits of this delightful gem of a book.
It is worth recalling the editors´ key recommendation, the reader "should keep in mind that a book on mathematics cannot be read quickly. Nor must he expect to understand all parts of the book on first reading. He should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subsequent remark". Finally they are perfectly right in stating that "The best way to learn mathematics is to do mathematics" and to that extent the problems set by the author are satisfactory.
The text niftily and succintly weaves some interesting history, beguiling mathematical problems and, in general, clear proofs of key theorems and assertions together with a smattering of problems to induce the reader to work out and understand the a well chosen set of related mathematical concepts. The books is an entertaining stroll which leads up to the solutions of the Pythagorean equation (for which integer values of the three sides are there right-angled triangles?) and to Fermat´s congruence (also known as the little Fermat´s theory) but pointing out important features on the way such as magic squares, Mersenne and Fermat primes, perfect and amicable numbers, numeration systems, the algebra of congruences and its application to checks on computations, finding out the day of the week any past or future date falls and how to build round-robin tournament schedules, amongst others. Written in 1967, it is a bit dated as regards some aspects of the field such as, understandably, the largest primes calculated using computers and I´m sure that, were Professor Ore (1899-1968) alive today and updating the book, he would carefully add a little something about the applications of number theory to cryptography. However, this does not detract from the merits of this delightful gem of a book.
It is worth recalling the editors´ key recommendation, the reader "should keep in mind that a book on mathematics cannot be read quickly. Nor must he expect to understand all parts of the book on first reading. He should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subsequent remark". Finally they are perfectly right in stating that "The best way to learn mathematics is to do mathematics" and to that extent the problems set by the author are satisfactory.
Displaying 1 - 2 of 2 reviews
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