Fibonacci Numbers

179 Pages·2002·4.824 MB·English

by  Nicolai N. Vorobiev (auth.)

Nicolai N. Vorobiev
iba aeei
um~ers
Translated from the Russian
by Mircea Martin
Springer Basel AG
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Nicolai N. Vorobiev
Russian Academy of Sciences
Leningrad-St. Petersburg
Originally published under the title of "Chisla Fibonacci" by Nauka, Moscow.
@ 1992 (6th edition) Nauka, Moscow
2000 Mathematics Subject Classification 00A05, 11839
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche 8ibliothek lists this publication in the Deutsche Nationalbibliografie; detailed
bibliographic data is available in the Internet at <http://dnb.ddb.de>.
This work is subject to copyright. AII rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, re-use of illustrations,
recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data
banks. For any kind of use permission of the copyright owner must be obtained.
Cover iIIustration: The idea for the cover picture has been taken from the article "Phyllotaxis
or Self-Similarity in Plant Morphogenesis" by F. Rothen (in: Fractals in Biology and Medicine,
ed. by T.F. Nonnenmacher, G.A. Losa, E.R. Weibel; Birkhăuser 1994).
The scales of the pineapple have been numbered from the bottom upward. They are regular
Iy disposed along spirals, in three different directions. The differences between neighbouring
scales in a spiral are constant, namely, 5, 8 and 13, respectively, and the Fibonacci sequence
begins with 1, 1,2,3, 5,8, 13,21,. ..
@ 2002 Springer Basel AG
Originally published by Birkhauser Verlag in 2002
Printed on acid-free paper produced from chlorine-free pulp. TCF
00
Cover design: Micha Lotrovsky, Therwil, Switzerland
ISBN 978-3-7643-6135-8 ISBN 978-3-0348-8107-4 (eBook)
DOI 10.1007/978-3-0348-8107-4
9 8 7 6 5 4 3 2 1 www.birkhauser-science.com
Contents
Foreword, Vll
Introduction, 1
1 The Simplest Properties of Fibonacci Numbers, 5
2 Number-Theoretic Properties of Fibonacci Numbers, 51
3 Fibonacci Numbers and Continued Fractions, 89
4 Fibonacci Numbers and Geometry, 125
5 Fibonacci Numbers and Search Theory, 149
Index, 175
Foreword
In elementary mathematics we encounter many challenging and interesting
problems, which are not connected with somebody's name but rather bear
the trait of a kind of "mathematical folklore." Such problems are scattered
throughout the existing popular, or purely recreational, mathematical liter
ature and often it is quite difficult to pinpoint the provenience of a specific
problem.
These problems frequently circulate in several versions. Sometimes a few
such problems come together as a single, more complex, problem; some other
times, on the contrary, a single problem is split up into a few simpler problems.
In other words, it is seldom possible to figure out where a particular problem
ends and another one begins. More to the point, we may think of each such
problem as a succinct mathematical theory with its own history, own top
ics, and own methods, by and large connected with the history, themes, and
methods of "great mathematics."
The theory of Fibonacci numbers is just one of this kind. Originating
from the famous Rabbit Problem, and going back in time about 770 years,
Fibonacci numbers provide one of the most captivating chapters of elementary
mathematics. Problems referring to Fibonacci numbers can be found in many
popular publications on mathematics, are studied at meetings of mathematical
school societies, and featured in mathematical competitions.
The present booklet addresses an array of issues which have served as
themes of several study sessions for the high school students enrolled in a
"mathematical circle" at the Leningrad State University. Granting the wishes
expressed by those who attended these meetings, the discussion mainly focused
on topics related to number theory, a subject developed in greater detail in
this text.
The first edition of this booklet was published in the early 1950's. Since
then a good many changes took place. First of all, and this is really important,
we witnessed a change in the level of mathematical skills and interests of the
traditional group of readers of popular mathematical books, which includes,
among others, high school students and their instructors alike. The newly cre
ated network of schools and classes, with highly specialized curricula in mathe
matics and physics, helped to considerably widen the mathematical horizon of
generation after generation of students and teachers who now are much more
interested in deep and complex results rather than in entertaining elementary
facts. Moreover, and this seems to be a fundamental feature of the history of
contemporary mathematics, the center of impact of mathematical research as
a whole underwent a striking relocation. For instance, number theory lost its
viii Foreword
paramount position and the interest in optimization problems gained a sud
den specific weight. Game theory arose as a new and self- determined branch
of mathematics. Numerical mathematics experienced an essential structural
change. Altogether, such mutations reshaped the subject matter of what one
refers to as popular literature on mathematics. Furthermore, Fibonacci num
bers showed up in the meantime in several mathematical issues. Among them
we wish to single out the proof due to Yu. V. Matiyasevich of the 10th Hilbert
Problem, as well as the less involved but widely publicized theory of search
for the extremum value of a unimodal function first developed apparently by
J. Kiefer.
Finally, an immense amount of previously unknown properties of Fi
bonacci numbers was established, and the interest in the study of these num
bers grew at a rapid rate. A lot of people from different countries, professional
or amateur mathematicians, supported the honorable hobby of "fibonaccism."
The most convincing evidence in this regard may be found in The Fibonacci
Quarterly, a journal published in the U.S.A. since 1963.
All these trends brought along several changes in the subject matter of
this book from one edition to another. In the present edition we included
a chapter on the Fibonacci search, as well as some related theoretical and
computational questions. Moreover, we added new topics on number theory to
Chapter 2, and this material served as useful information for proving the Tenth
Hilbert Problem. In addition, we "pushed" to the general level the contents
of Chapters 3 and 4. In Chapter 3 we present a few classical theorems that
address the accuracy of approximating real numbers by continued fractions,
and describe the role of Fibonacci numbers in this regard. Chapter 4 includes
an analysis of the game "tzyan-she-tzy" in the broader perspective of game
theory, that relies on a detailed investigation of the Fibonacci representation
of natural numbers. Finally, we responded to the request to complement the
book with some specific material illustrating connections between Fibonacci
numbers and topics in computer science. This has been done by including in
Chapter 1 a general presentation of the implementation and uses in computer
technology of the Fibonacci numeration system, as well as its refined analog,
the "golden" numeration system.
The book, as before, does not presuppose that the reader's knowledge
extends beyond the average limits of high school courses. The more difficult
parts dealing with the issues mentioned above could be skipped at a first
reading with no harm to the understanding of the remaining material.
I took advantage of an amiable observation made by Professor N. 1. Feld
man from Moscow, who pointed out a logical oversight at the end of Chapter 3
in the previous editions that concerned the special status of the limiting value
of ratios of neighboring Fibonacci numbers in approximating real numbers
Foreword ix
by continued fractions. His constructive advice helped me to get rid of this
omission. This just makes it more pleasing for me to express my gratitude to
Naum Ilich who a long time ago in the academic year 1940-1941, as head of a
mathematical circle at the Leningrad State Univesity, strove to convey to me,
a high school student, the skills of number-theoretic reasonings.
Vyritza, 1989
N.N. VOROBIEV
Translator's Note: I would like to thank my wife, Larisa Martin, for typeset
TEX,
ting the manuscript in Merrie Skaggs for checking the accuracy of the
translation, and Dr. Thomas Hintermann for constant support and assistance.
To them all lowe a debt of gratitude.
MIRCEA MARTIN
Baker University
Introduction
1. The early history of mathematics abounds in accounts of prominent mathe
maticians. Many of the accomplishments of ancient mathematics are yet highly
regarded for the sharpness of mind of their authors, and the names of Euclid,
Archimedes, and Heron are nowadays known to every educated person.
Things look quite different when we check over the mathematics of the
Middle Ages. Except for Viete, who actually lived in the sixteenth century, and
apart from mathematicians much closer in time to us, current school courses of
mathematics do not record any single name connected with the Middle Ages.
As might be expected, this is not an accident. During that epoch, mathematics
developed extremely slowly and there were very few notable mathematicians.
The greater then should be our interest in the work Liber abaci - "A book
on abacus" - written by an outstanding Italian mathematician, Leonardo of
Pisa , who is better known by his nickname Fibonacci, short for filius Bonacci,
which means son of Bonacci. This famous book, published in 1202, is known
to us through a second edition that appeared in 1228.
Liber abaci is a voluminous compendium including almost all the arith
metical and algebraic knowledge of those times. The book played an important
part in the development of mathematics in Western Europe through many sub
sequent centuries. In particular, it was from this book that Europeans became
acquainted with the Hindu-Arabic numerals.
The material collected in Liber abaci is illustrated by a large number of
problems that makes up a significant part of the treatise.
We confine ourselves to presenting one of Fibonacci's problems, known as
the Rabbit Problem. It can be found on pages 123-124 of the 1228 manuscript.
"How many pairs of rabbits can be bred from a single pair in one year?"
"Someone put a pair of rabbits in a certain place, entirely surrounded
by a wall, to find out how many pairs of rabbits can be bred from it in one
year. The nature of these rabbits is such that every month a pair of rabbits
produces another pair, and rabbits start breeding in the second month after
their birth.
As the first pair breeds a pair in the first month, then duplicate it and
there will be 2 pairs in a month. From these pairs, one - namely the first -
gives birth in the following month to another pair, and thus there are 3 pairs
in the second month. From these three pairs in one month two will become
pregnant, so that two more pairs of rabbits will be born in the third month,
and the number of pairs will reach 5. From these in the same month three will
be pregnant, so that there will be 8 pairs of rabbits in the fourth month. From
these pairs five will breed five other pairs, which added to the eight pairs will
N. N. Vorobiew (ed.), Fibonacci Numbers
© Birkhäuser Verlag 2002
2 Introduction
give 13 pairs in the fifth month. From these the five pairs which
Pair were born during the fifth month will not conceive in that month,
1 but the other eight will, so that there will be 21 pairs in the sixth
month. Adding these to the 13 pairs that will be born in the next
month, there will be 34 pairs in the seventh month. Adding these
First
to the 21 pairs born in the next month, gives 55 pairs in the eighth
2
month. Adding these to the 34 pairs born in the next month, gives
89 pairs in the ninth month. Adding these to the 55 pairs born
Second
in the next month, gives 144 pairs in the tenth month. Adding
3
these to the 89 pairs born in the next month, gives 233 pairs in
the eleventh month. To these we finally add the 144 pairs born
Third
next month and so there will be 377 pairs in the twelfth month.
5
This is the number of pairs produced from the first pair at the
given place by the end of one year.
Fourth
You can see in the margin how we have done this, namely by
8
adding the first number to the second, hence 1 and 2, the second
to the third, the third to the fourth, the fourth to the fifth, and
Fifth so on, one after another, until the tenth is added to the eleventh,
13 hence 144 and 233, and so we got the total number of pairs of
rabbits, namely 377, and in this way you can do it for the case of
Sixth an indefinite numbers of months."
21
2. We now pass from rabbits to numbers and examine the follow
ing numerical sequence
Seventh
34
(1)
Eighth in which every term equals the sum of the two preceding terms,
55 that is,
(2)
Ninth
for any n > 2.
89
Such sequences, in which every term is defined as a function
of previous terms, are often encountered in mathematics and are
Tenth
called recurrent sequences . The process of consecutively comput
144
ing terms of a recurrent sequence is called a recurrence process,
and the specific equation that describes a recurrence process, like
Eleventh equation (2) above, is called a recurrence relation. The reader can
233
find additional details concerning the general theory of recurrent
sequences in the already mentioned book by A. I. Markushevich
Twelfth (see page iv).
377 A first point we want to make is that the recurrence rela
tion (2) by itself does not enable us to calculate the terms of
Introduction 3
sequence (1). We can form infinitely many different numerical sequences sat
isfying that condition, such as,
2, 5, 7, 12, 19, 31, 50, ... ,
1, 3, 4, 7, 11, 18, 29, ... ,
-1, -5, -6, -11, -17, ... ,
and so on.
Therefore, whenever the uniqueness of sequence (1) is required, condition
(2) is obviously insufficient, so we must impose certain additional conditions.
For example, we can indicate the first few terms of the sequence. How many
initial terms of sequence (1) should be listed so that the computation of all its
subsequent terms is possible by employing condition (2) alone?
To answer this question we notice that not each term of sequence (1) can
be obtained from (2) if only because not all the terms of (1) have two preceding
terms; for instance, the first term of the sequence has no preceding terms at
all, and the second term is preceded by just one term. It follows that besides
condition (2) we must know the first two terms of the sequence in order to
produce it.
This will actually suffice in allowing us to compute all the other terms of
sequence (1). Indeed, U3 can be calculated as the sum of the two prescribed
terms, Ul and U2; next, U4 equals the sum of U2 and the previously calculated
U3; further, U5 is the sum of the two already calculated U3 and U4, and so on,
"for the case of an indefinite number of terms." In this way, by merely passing
from two consecutive terms to their immediately following term, we can reach
every term with a prescribed subscript, and calculate it.
3. We now turn our attention to the important particular case of sequence
(1) subject to condition (2), where Ul = 1 and U2 = 1. As we just pointed out
above, condition (2) allows us to successively calculate all the other terms of
this sequence. It is easy to verify that in this case the first fourteen terms are
the numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
which we met before in the Rabbit Problem. In honor of the author of this
problem, the recurrent sequence (1), when Ul = U2 = 1 and the recurrence
relation is given by equation (2), is called the Fibonacci sequence ,and its
terms are referred to as Fibonacci numbers .
Fibonacci numbers enjoy a great deal of both interesting and important
properties, and this booklet is entirely devoted to their investigation.