Lectures on Number Theory

286 Pages·1986·5.56 MB·English

by  Nikolaos Kritikos (auth.)

Universitext
Editors
F.W. Gehring
PR. Halmos
C.C. Moore
Universitext
Editors: F.W. Gehring, P.R. Halmos, C.C. Moore
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Lectures on
Number Theory
Presented by
Adolf Hurwitz
Edited for Publication by
Nikolaos Kritikos
Translated, with some additional material, by
William C. Schulz
Springer Verlag
New York Berlin Heidelberg Tokyo
Nikolaos Kritikos William Schulz (Translator)
Parnithos 48 Northern Arizona University
154 52 Psychiko Department of Mathematics
Athens Flagstaff, AZ 86011
Greece U.S.A.
AMS Classifications: 10-01, IOA05, IOAI5, IOA32, IOC99
Library of Congress Cataloging in Publication Data
Hurwitz, Adolf
Lectures on number theory.
(Universitext)
Translated from the German.
Bibliography: p.
Includes index.
I. Numbers, Theory of. I. Kritikos, Nikolaos,
1894- . II. Title.
QA24l.H85 1986 512.77 85-25093
© 1986 by Springer-Verlag New York Inc.
of
Softcoverreprint the hardcover 1st edition 1986
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New
York 10010, U.S.A.
9 8 7 6 5 4 3 2 1
ISBN-13: 978-0-387-96236-8 e-ISBN-13:978-1-4612-4888-0
DOl: 10.107/978-1-4612-4888-0
Preface
During the academic year 1916-1917 I had the good fortune to be a student
of the great mathematician and distinguished teacher Adolf Hurwitz, and to
attend his lectures on the Theory of Functions at the Polytechnic Institute of
Zurich. After his death in 1919 there fell into my hands a set of notes on
the Theory of numbers, which he had delivered at the Polytechnic Institute.
This set of notes I revised and gave to Mrs. Ferentinou-Nicolacopoulou with a
request that she read it and make relevant observations. This she did
willingly and effectively. I now take advantage of these few lines to express
to her my warmest thanks.
Athens, November 1984
N. Kritikos
About the Authors
ADOLF HURWITZ was born in 1859 at Hildesheim, Germany, where he attended
the Gymnasium. He studied Mathematics at the Munich Technical University and
at the University of Berlin, where he took courses from Kummer, Weierstrass
and Kronecker. Taking his Ph.D. under Felix Klein in Leipzig in 1880 with a
thes i s on modul ar funct ions, he became Pri vatdozent at Gcitt i ngen in 1882 and
became an extraordinary Professor at the University of Konigsberg, where he
became acquainted with D. Hilbert and H. Minkowski, who remained lifelong
friends. He was at Konigsberg until 1892 when he accepted Frobenius' chair at
the Polytechnic Institute in Z~rich (E.T.H.) where he remained the rest of his
1 i fe.
Hurwitz's mathematics was heavily influenced by Felix Klein. He worked
mainly in number theory and related areas of complex analysis, including
modular functions, Riemann surfaces, and complex multiplication. Hurwitz
originated the invariant volume for integration on the orthogonal groups,
which was later generalized to Haar measure on topological groups. He showed
that the real numbers, complex numbers, quaternions and Cayley octaves are the
only algebras without divisors of zero and with quadratic norm over the real
numbers. This result became one of the pillars of the theory of algebras.
Hurwitz did pioneering work on the arithmetic of quaternions, and discovered
the most fruitful definition of an "integral" quaternion. Also interesting
are his papers on various aspects of continued fractions.
Hurwitz died in Z~rich in 1919.
viii
NIKOLAOS KRITIKOS was born of Greek parents in Constantinople in 1894.
He studied Mathematics at the Universities of Athens, Gcittingen and Zu"rich,
and at the Polytechnic Institute (E.T.H.) in Zu"rich. Among his teachers were
C. Caratheodory and Adolf Hurwitz. He received his Dr. Phil. from the
Philosophical Faculty II of the University of Z~rich in 1920 with a
dissertation written under Dr. G. PIo lya. He became full Professor of Higher
Mathematics at the University of Thessaloniki in 1928. During the years
1933-1946 and 1951-1963 he served as full Professor of Higher Mathematics at
the E.M. Polytechnic Institute in Athens.
Translator's Preface
This English version of A. Hurwitz's Lectures on Number Theory has been
taken from the edited version of Prof. N. Kritikos, with occasional
consultation of the original notes. A very few modifications have been
incorporated into the last chapter of the English version to take advantage of
the greater familiarity of present day students with matrices.
The translator wishes to point out the splendid organization used by
Prof. Hurwitz. For example, in the last chapter the theory of binary
quadratic forms and the theory of continued fractions are developed together
in about the same space which would be necessary to develop the theory of
continued fractions alone. Another example is the formula (40.1) derived from
Gauss' lemma, from which follows the law of quadratic reciprocity and both
complementary theorems.
In order to make the book more useful as a classroom text, the translator
has added problems at the end of the chapters. These are of three types. The
first, numerical examples, have been constructed with the aim of providing
insight into the general situation with the least amount of calculation. The
second class of problems provides computational algorithms for the theoretical
material covered in the book. The third class of problems attempts to provide
interesting extensions of the theory in the main text.
The main text is completely independent of the problems. Problems are
not organized by degree of difficulty. Rather, an attempt has been made to
roughly correlate the problems with the sections of the text, so that they may
be worked as the chapter is read. The problems are for the most part easy,
and copious hints have been provided, so that they may be solved in a
reasonable time.
x
The translator would like to thank Prof. Kritikos for his active
cooperation in the correction of the English text. He would also like to
thank Evelyn Wong and Kim Poole of the Ralph M. Bilby Research Center for
their fine job of typing, and Northern Arizona University for its support of
the entire project, and my wife Maria M. Schulz for her generous contributions
of time and effort on the project.
The translator would greatly appreciate it if any errors detected by
readers are forwarded to him, as well as suggestions for additional problems
or improvement in the existing ones.
William C. Schulz
Mathematics Oepartment
Northern Arizona University
Flagstaff, Arizona USA 86011
Table of Contents
CHAPTER 1. BASIC CONCEPTS AND PROPOSITIONS
1. The Principle of Descent •••••••••••••••••••••••••••••••••••••••••••••• 1
2. Divisibility and the Division Algorithm ••••••••••••••••••••••••••••••• 3
3. Prime Numbers ••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 6
4. Analysis of a Composite Number into a Product of Primes ••••••••••••••• 8
5. Divisors of a Natural Number n, Perfect Numbers •••••••••••••••••••••• 12
6. Common Divisors and Common Multiples of two or more Natural Numbers •• 15
7. An Alternate Foundation of the Theory of
The Greatest Common Divisor ....................................... 18
8. Euclidean Algorithm for the G.C.D. of two Natural Numbers •••••••••••• 21
9. Relatively Prime Natural Numbers ••••••••••••••••••••••••••••••••••••• 23
10. Applications of the Preceding Theorems ••••••••••••••••••••••••••••••• 26
11. The Function ~(n) of Euler ••••••••••••••••••••••••••••••••••••••••••• 32
12. Distribution of the Prime Numbers in the Sequence
of Natural Numbers •••••••••••••••••••••••••••••••••••••••••••••••• 37
Problems for Chapter 1 ••••••••••••••••••••••••••••••••••••••••••••••• 45
CHAPTER 2. CONGRUENCES
13. The Concept of Congruence and Basic Properties ••••••••••••••••••••••• 51
14. Criteria of Divisibility ••••••••••••••••••••••••••••••••••••••••••••• 53
15. Further Theorems on Congruences •••••••••••••••••••••••••••••••••••••• 56
16. Residue Classes mod m• ••••••••••••••••••••••••••••••••••••••••••••••• S8
17. The Theorem of Fermat •••••••••••••••••••••••••••••••••••••••••••••••• 6D
18. Generalized Theorem of Fermat •••••••••••••••••••••••••••••••••••••••• 61