(LISC Press-WSPC) A K Mallik - The Story of Numbers-World Scientific (2017) (1).pdf

The Story
of Numbers
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IIScPress-WSPC Publication
Print ISSN: 2529-7864
Online ISSN: 2529-7872
Series Editor: Amaresh Chakrabarti (Indian Institute of Science, India)
Published:
Vol. 1: Remembering Sir J C Bose
by D P Sen Gupta, M H Engineer & V A Shepherd
Vol. 2: Climbing the Limitless Ladder: A Life in Chemistry
by C N R Rao
Vol. 3: The Story of Numbers
by Asok Kumar Mallik
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NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
World Scientiﬁc
Asok Kumar Mallik
Asok Kumar Mallik
Indian Institute of Engineering Science and Technology, Shibpur, India
Indian Institute of Engineering Science and Technology, Shibpur, India
The Story
The Story
of Numbers
of Numbers
of Numbers
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Library of Congress Cataloging-in-Publication Data
Names: Mallik, A. K. (Asok Kumar), 1947–
Title: The story of numbers / by Asok Kumar Mallik
(Indian Institute of Engineering Science and Technology, India).
Description: New Jersey : World Scientific, 2017. | Series: IIScPress-WSPC publication ; vol 3 |
Includes bibliographical references.
Identifiers: LCCN 2017013738 | ISBN 9789813222922 (hc : alk. paper)
Subjects: LCSH: Numeration--History. | Mathematics--History | Mathematics--Philosophy.
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To the loving memory of my wife Kaberi
v

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Preface
This is not a textbook of mathematics, neither written by a
mathematician nor written for mathematicians. There is a classical
text entitled “Numbers”, which was originally written in German and
subsequently translated into English. Both versions were published
by Springer in the series on Graduate Texts in Mathematics. In the
preface of the English edition, it has been mentioned that “this is not
a book for the faint-hearted”. The present book is mainly aimed at
those “faint-hearted” people who may otherwise have some interest
in mathematics as non-professionals. I and, of course, the publisher
think that there are enough people in this category so that a book
at this level is desired. The present book can be followed with the
knowledge of high-school level mathematics and is thus suitable for
school students as well.
An attempt has been made to arrange some basic facts on
various kinds of numbers that mathematicians have found over a
period of a few millennia. Mathematics demands rigorous treatments
but here the emphasis is more on stories and anecdotes rather
than rigour. The beauty of surprising connections between various
numbers originating in different contexts, curious properties and
patterns exhibited by innocuous numbers, how some new kinds of
numbers have been created — all these are discussed more in the
form of a story rather than hardcore mathematics.
vii

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viii Preface
Mathematics normally is not perceived as a popular subject.
Therefore, I do not dare to say that this is targeted as a popular
book on mathematics. Popular books on mathematics may be more
unpopular. However, I have included a long list of excellent references
which are mostly in the category of so-called popular books, rather
than textbooks, on mathematics. The reader may consult these
references to dig deeper into any specific topic of his/her choice.
Another very rich source of interesting information on “Numbers” are
the set of video lectures in the series “Numberphile”, freely available
over the Internet in youtube. A few references are also mentioned
as footnotes. These are the sources of the specific information under
discussion at that location and are not meant for further reading.
Many friends have helped me to complete this book. Special
mention must be made of Professors Raminder Singh, Aparna (Dey)
Ghosh, Pradipta Bandyopadhyay, Anindya Chatterjee and G. K.
Ananthsuresh, all of whom read different versions of the manuscript
and suggested a lot of corrections and improvements. For the
mistakes that still remain, only I am responsible. My young friends
Amit, Suhas and Rajarshi always encouraged me with their frank
assessments of the contents and the quality of writing. Amit especially
helped me by supplying a lot of suitable material and providing
me a tutorial on p-adic numbers, which I have finally included on
his insistence. Finally I thank Professor Goutam Bandyopadhyay for
allowing me to use his notes on hyperreal numbers. I express my
sincere gratitude to all these people. Hope some non-mathematician
adults and school children enjoy reading the book.

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About the Author
Asok Kumar Mallik is Honorary Dis-
tinguished Professor at Indian Institute
of Engineering Science and Technology,
Shibpur. During 1971–2009 he was in
the Faculty of Mechanical Engineering at
Indian Institute of Technology, Kanpur. He
was a Commonwealth Scholar at The Insti-
tute of Sound and Vibration Research at
Southampton, England and an Alexander
von Humboldt Fellow at TH, Aachen and
TU, Darmstadt in Germany. He has been
elected a Fellow of The Indian National Academy of Engineering and
all the three Science Academies of India. He has authored one book
and several articles on high school mathematics.
ix

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Contents
Preface vii
About the Author ix
1. Introduction 1
2. Integers 11
2.1 Representation of Integers . . . . . . . . . . . . 11
2.2 Curious Patterns in Numbers . . . . . . . . . . 12
2.2.1 Multiplication . . . . . . . . . . . . . . 13
2.2.2 Combination of multiplication and
addition . . . . . . . . . . . . . . . . . 13
2.2.3 Consecutive integers . . . . . . . . . . 14
2.2.4 Pascal’s triangle . . . . . . . . . . . . . 16
2.3 Iterations . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Number of even, odd and total digits . 18
2.3.2 Sum of squares of the digits . . . . . . 18
2.3.3 Sum of cubes of the digits . . . . . . . 19
2.3.4 A ﬁxed point at 1089 . . . . . . . . . . 20
2.3.5 Kaprekar numbers . . . . . . . . . . . 20
2.3.6 Collatz conjecture (hailstone numbers) 21
2.4 Prime Numbers . . . . . . . . . . . . . . . . . . 22
2.4.1 Euclidean primes . . . . . . . . . . . . 24
2.4.2 Mersenne primes . . . . . . . . . . . . 24
xi

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xii Contents
2.4.3 Double Mersenne primes . . . . . . . . 26
2.4.4 Fermat primes . . . . . . . . . . . . . 26
2.4.5 Pierpoint primes . . . . . . . . . . . . 28
2.4.6 Sophie Germain primes . . . . . . . . 28
2.4.7 Pillai primes . . . . . . . . . . . . . . . 29
2.4.8 Ramanujan primes . . . . . . . . . . . 30
2.4.9 Wilson primes . . . . . . . . . . . . . . 30
2.4.10 Twin primes . . . . . . . . . . . . . . . 30
2.4.11 Carmichael numbers . . . . . . . . . . 32
2.4.12 “emirp” . . . . . . . . . . . . . . . . . 33
2.4.13 Cyclic primes . . . . . . . . . . . . . . 33
2.4.14 Prime digit/composite digit primes . . . 34
2.4.15 Almost-all-even-digits primes . . . . . 34
2.4.16 Palindromic and plateau primes . . . . 34
2.4.17 Snowball primes . . . . . . . . . . . . 35
2.4.18 Russian Doll primes . . . . . . . . . . . 35
2.4.19 Pandigital primes . . . . . . . . . . . . 35
2.4.20 Very large prime numbers with
repeated pattern . . . . . . . . . . . . 35
2.4.21 Miscellany . . . . . . . . . . . . . . . . 36
2.5 Composite Numbers . . . . . . . . . . . . . . . 38
2.5.1 Highly composite numbers . . . . . . . 38
2.5.2 Sierpinski’s numbers . . . . . . . . . . 39
2.5.3 Perfect and associated numbers . . . . 40
2.5.4 Friendly (Amicable) numbers . . . . . . 42
2.5.5 Sociable numbers . . . . . . . . . . . . 43
2.5.6 Untouchable numbers . . . . . . . . . 43
2.5.7 Smith numbers . . . . . . . . . . . . . 43
2.6 Sequences . . . . . . . . . . . . . . . . . . . . 44
2.6.1 Fibonacci (Hemachandra) sequence . . 44
2.6.2 Padovan sequence . . . . . . . . . . . 47
2.6.3 Perrin sequence . . . . . . . . . . . . . 48
2.6.4 Look-and-say sequence . . . . . . . . 49
2.7 Pythagorean Triples (Triplets) . . . . . . . . . . . 50
2.8 Taxicab and Similar Numbers . . . . . . . . . . 52
2.8.1 Taxicab numbers . . . . . . . . . . . . 52
2.8.2 Numbers refuting Euler’s conjecture . . 55

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Contents xiii
2.9 Narcissistic and Similar Numbers . . . . . . . . 56
2.9.1 Narcissistic numbers . . . . . . . . . . 56
2.9.2 Factorians and factorial loops . . . . . 57
2.9.3 Kaprekar numbers . . . . . . . . . . . 58
2.9.4 SP and S + P numbers . . . . . . . . . 59
2.10 Some Unassuming Integers . . . . . . . . . . . 59
2.10.1 Integer 4 . . . . . . . . . . . . . . . . 59
2.10.2 Integer 7 . . . . . . . . . . . . . . . . 60
2.10.3 Integers 9, 23, 239 . . . . . . . . . . . 60
2.10.4 Integers 24 and 70 . . . . . . . . . . . 60
2.10.5 Integer 26 . . . . . . . . . . . . . . . . 61
2.10.6 Integer 77 . . . . . . . . . . . . . . . . 61
2.11 Very Large Numbers . . . . . . . . . . . . . . . 62
2.11.1 Ogha, Mahaugha, googol and
googolplex . . . . . . . . . . . . . . . 63
2.11.2 Measurable inﬁnity . . . . . . . . . . . 64
2.11.3 Colour combinations of a Rubik cube . 64
2.11.4 Archimedes cattle problem . . . . . . . 65
2.11.5 Skewes’s number . . . . . . . . . . . . 65
2.11.6 Moser number and Graham number . 66
3. Real Numbers 69
3.1 Rational Numbers . . . . . . . . . . . . . . . . 69
3.2 Irrational Numbers . . . . . . . . . . . . . . . . 70
3.3 Transcendental Numbers . . . . . . . . . . . . 71
3.4 Decimal Representation . . . . . . . . . . . . . 73
3.5 Continued Fraction Representation . . . . . . . 75
3.6 Iterations . . . . . . . . . . . . . . . . . . . . . 78
3.6.1 Square root — Babylonian . . . . . . . 81
3.6.2 Square root — Indian . . . . . . . . . 82
3.6.3 Gauss’s constant . . . . . . . . . . . . 83
3.6.4 Logistic map and Feigenbaum numbers 84
3.7 Special Rational Numbers . . . . . . . . . . . . 87
3.7.1 Unique Egyptian fractions with sum unity 87
3.7.2 A Steinhaus problem . . . . . . . . . . 87
3.7.3 Parasite numbers . . . . . . . . . . . . 88
3.7.4 Congruent numbers . . . . . . . . . . 89

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xiv Contents
3.7.5 Bernoulli numbers . . . . . . . . . . . 90
3.7.6 Curious periodic patterns . . . . . . . 92
3.8 Special Irrational and Transcendental
Numbers . . . . . . . . . . . . . . . . . . . . . 93
3.8.1 Pythagoras’s number:
√
2 . . . . . . . 94
3.8.2 Golden sections . . . . . . . . . . . . . 94
3.8.3 Vishwanath number . . . . . . . . . . 97
3.8.4 Schizophrenic numbers . . . . . . . . . 98
3.8.5 Oldest universal mathematical
constant π . . . . . . . . . . . . . . . . 99
3.8.6 Base of natural logarithm e . . . . . . 103
3.8.7 Famous formulas having both
π and e . . . . . . . . . . . . . . . . . 108
3.8.8 Apery’s constant . . . . . . . . . . . . 111
3.8.9 Euler’s constant γ . . . . . . . . . . . . 112
3.8.10 Liouville’s number . . . . . . . . . . . 114
3.8.11 Champernowne’s number . . . . . . . 114
3.8.12 Hilbert’s number . . . . . . . . . . . . 114
4. Imaginary and Complex Numbers 115
4.1 A Brief Early History of
√
−1 . . . . . . . . . . . 117
4.2 Geometric Representation of Complex
Numbers . . . . . . . . . . . . . . . . . . . . . 119
4.3 Euler’s Fabulous Formula . . . . . . . . . . . . 121
4.4 Complex Exponentiation and Special
Numbers . . . . . . . . . . . . . . . . . . . . . 123
4.5 Fundamental Theorem of Algebra . . . . . . . . 125
4.6 Gaussian Integers and Gaussian Primes . . . . 126
4.7 Riemann Hypothesis . . . . . . . . . . . . . . . 129
4.8 Iterations . . . . . . . . . . . . . . . . . . . . . 134
5. Special Numbers 143
5.1 Hyperreal Numbers . . . . . . . . . . . . . . . 143
5.2 Quaternions . . . . . . . . . . . . . . . . . . . 146
5.3 Dual Numbers . . . . . . . . . . . . . . . . . . 149
5.3.1 Arithmetic . . . . . . . . . . . . . . . . 149
5.3.2 Function . . . . . . . . . . . . . . . . . 150

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Contents xv
5.3.3 Dual angle and trigonometry . . . . . 150
5.3.4 Trigonometry . . . . . . . . . . . . . . 151
5.4 p-Adic Numbers . . . . . . . . . . . . . . . . . 152
5.4.1 Decimal representation of real numbers
and 10-adic numbers . . . . . . . . . 153
5.4.2 p-Adic integers . . . . . . . . . . . . . 154
Appendix A 161
A.1 Solution of Equation (1.2) . . . . . . . . . . . . 161
A.2 Brahmagupta’s Equation and Its Solution . . . . 161
A.3 Solution to Equation (2.26) . . . . . . . . . . . 164
Appendix B 165
B.1 Sum of Integral Powers of Natural Numbers . . 165
Appendix C 169
C.1 Origin of Curious Patterns (Section 3.7.6) . . . 169
References 171
Index 175

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Chapter 1
Introduction
Mathematics is a grand edifice of the human intellect constructed
over a period of a few millennia, cutting across different civilisa-
tions. In natural and biological sciences, old theories make way
for new ones, but in mathematics, new developments are added,
sometimes connecting previous separate areas, without replacing
them. Freeman Dyson in a lecture classified mathematicians into
two groups, namely, birds and frogs. Birds take an overview of
different areas and establish connections among them, whereas frogs
continue to find new gems in one restricted area. The activities of
both groups have enriched the subject. The ever increasing field
of modern mathematics starts with counting. For counting we use
positive integers 1, 2, 3, . . . , which are aptly called natural numbers.
Everybody notices one Sun, two eyes, five fingers and so on in nature.
Leopold Kronecker said, “God made the natural numbers, everything
else is man’s handiwork” [9].
The sense of quantity of discrete objects (elements in a set) in terms
of one, two, few and many is also possessed by some animals, birds
and insects. Interesting experiments conducted on these species have
confirmed that crows can keep a count of up to three [6]. In one such
experiment, first, one person was sent up a tree where a crow couple
was taking care of their offspring. The couple flew away to a nearby
1

2 The Story of Numbers
tree and kept watching their nest. They returned as soon as the person
came down. Then two persons were sent up one after another. When
the first person came down, the crows waited and returned only after
the second person came down after sometime. Then three persons
went up one after another and came down one after another with
some time gap. The crows waited until all the persons came down.
But with more than three persons sent up the tree one after another,
the crows returned as soon as the third person came down implying
that now they had lost count.
It has been confirmed that parrots, squirrels and chimps can be
trained to keep a count of up to six. Pickover [27] mentions desert
ants in Sahara that appear to have a built-in “computer” that counts
their steps as they go out to a fairly large distance for collecting food.
They bring back the collected food to their nests. Experiments were
conducted by shortening and lengthening their strides after the food
was collected. Towards this end, the leg lengths of the ants were
manipulated, shortened by amputating and lengthened by adding
stilts. With shortened strides it was noticed that the ants started search-
ing for their nests way before reaching the destination. On the other
hand, with longer strides they continued to move, going past their
nests. Thus it appears that the process of counting in some form in
terms of natural numbers exists in nature without human intervention.
One of the greatest advances in mathematics took place when
the domain of positive integers, or natural numbers, was extended
by introducing another integer as zero (with the current symbol 0).
A vast literature exists on the history and significance of zero [17,
33]. Without getting into the intricacies of this great idea, we just
notice that 0 is used both as an even number and a digit. As implied
in the title of reference [17], zero as a number means “nothing”.
Counting starts from 1 and 0 signifies absence. In a lighter vein,
we refer to the following story of a kindergarten kid. When asked
to name five animals of Africa, the kid answered three lions and
two elephants. The teacher was not satisfied and said, “You have not
named five animals.” The stubborn kid continued: “OK — three lions,
two elephants, zero zebras, zero giraffe and zero deer and now I have
named five animals”! Mathematically, the new answer just suggests
that zebra, giraffe and deer are not to be found in Africa.

Introduction 3
Ancient Indian mathematicians are credited for the invention of
zero and the introduction of the now universally accepted Hindu–
Arabic system of writing numbers which is so much superior to other
systems that were used in different civilisations. The Indian math-
ematician Sridhara laid down the rules for arithmetical operations
involving the number zero.
Natural numbers are unending or limitless. In mathematics, we
use a symbol ∞ to express the unlimited infinity. This symbol was
first used by Wallis in 1655 and mathematicians have used it ever
since. Just as on zero, a vast literature also exists related to this
abstract mathematical idea [8]. Even religious mysticism is connected
to this mathematical concept of infinity [12]. Infinities appear in
different contexts in different areas of mathematics. It has also been
established that not all infinities are of the same order and there
exist methods to compare different orders of infinities. Again without
going into any further details, we now describe how zero and infinity
are literally opposite poles in the theory of stereographic projections
developed by Riemann [33]. Imagine a translucent sphere placed on
an infinite horizontal plane. A point light source is placed at the top
most point (the North Pole) of the sphere. Every point on the surface
of the sphere casts its shadow on the horizontal plane. There exists
a one-to-one correspondence between a point on the sphere and its
shadow. The shadow of the bottom most point (the South Pole) is at
the coincident point with the plane, i.e., the point of contact between
the sphere and the plane. We consider this point as the origin or zero.
The shadows of the points which are near the light source are very
far off from this origin. The point corresponding to all the points at
infinity (in every direction) is the North Pole. Thus the zero and the
infinity are literally the opposite poles, zero at the South Pole and
infinity at the North Pole.
It is mentioned in “The Book of Numbers”a that many math-
ematicians, especially those working with infinities, prefer to start
counting from zero rather than one. For example, Cantor did it while
aJ. H. Conway and R. K. Guy (1995). The Book of Numbers. New York: Copernicus,
Springer-Verlag.

devising methods of counting different infinities. Automatic counting
machines, unless otherwise taught, also start counting from zero.
A three-digit ticket dispenser shows first 1000 tickets numbered as
000 to 999. The digital display of an odometer (used in a car) shows
00000 to 99999 to indicate the beginnings of the first to hundred
thousandth kilometre.
Starting from non-negative integers, using one or more of the
direct (like addition, multiplication, exponentiation) and/or inverse
(like subtraction, division, root extraction) arithmetical operations
on these, different types of other numbers have been created.
These “unnatural” numbers, not to be found anywhere in nature,
are abstract products of the human imagination. For example,
negative integers −1, −2, −3, . . . have been created to make sense of
subtracting a bigger positive integer from a smaller one. Arithmetic
rules for handling positive and negative numbers were laid down
by the Indian mathematician Brahmagupta. Of course, a negative
number signifying the quantity of a natural object, like the number of
bees, was regarded as meaningless. But in one instance, a negative
distance was interpreted as a distance measured in the opposite
direction.
Besides negative integers, these man-made numbers include
rational, irrational, transcendental, imaginary, complex, dual, hyper-
real, p-adic numbers and quaternions, and so on. Mathematicians
unravel the mysteries of all such numbers. They also enjoy deci-
phering special characteristics of certain numbers in each category.
The language of mathematics, created by using different types of
numbers, has been most profitably used by natural scientists. They
have used this language to express their ideas clearly, concisely and
quantitatively. Famous physicist Wigner was surprised at this “unrea-
sonable success of (man-made) mathematics towards describing and
understanding the working of nature”.
In this short story of numbers, we discuss various types of numbers
and mention special types and characteristics of some well-known
and some not-so-well-known numbers. The objective is not to discuss
serious mathematics, but point out curious things about numbers
which are expected to be recreational for lay persons (amateurs)

Introduction 5
having an interest in something mathematical. High school students,
who regularly use numbers, may find some new and interesting
information which is not available in textbooks of mathematics.
It must be admitted that serious mathematics requires rigorous
treatment of every concept. Even natural numbers have Dedekind’s
set-theoretic and Peano’s axiomatic definitions and the naïve set
theory may not suffice. The development of every concept follows
a strict logical sequence. But sometimes this process is abandoned.
In the introduction of a classical text in mathematics, entitled “Num-
bers”,b it was conceded that logically the last chapter of the book
should have been the first chapter. But the decision was to follow an
age-old advice of not to begin at the beginning, as it is always the
most difficult part. About rigour in mathematics, sometimes famous
mathematicians also make jokes. It is said that too much rigour
may cause rigor mortis to set in stopping all movements. To quote
V. I. Arnold, “mathematicians do not understand a sentence like ‘Bob
washed his hands’. They would rather simply say, ‘There exists a time
interval (t1, 0) in which the natural mapping t → Bob(t) represents
a set of people with dirty hands and there is another interval [0, t2)
where the same mapping denotes a set complementary to the one
considered earlier.’ ”
Jokes apart, it may be mentioned at this stage, that numbers were
not always at the centre of all of mathematics. Greek geometers, like
Pythagoreans, founded their mathematics on the abstract concepts of
points and lines. They were interested in investigating the properties
of shapes and sizes. Numbers were used to express various measures
like the length of a line or the area enclosed by a geometrical figure
or the volume of a solid.
Geometry was held in very high esteem by great minds. Galileo
said that the secrets of nature are written in the language of
geometry. At the entrance of Plato’s academy it was inscribed “Let
no one ignorant of geometry enter here”. In a late nineteenth
bH. D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukrich,
A. Prestel and R. Remmert (1991). Numbers. New York: Springer.

century senate meeting of Yale University, some members were
criticising replacement of courses in classics and languages by those
in mathematics and science. Gibbs, a man of few words, got up
and said “Gentlemen what are we discussing? Mathematics is a
language.” However, it may also be mentioned that not all great
men were so kind to mathematics and mathematicians. Goethe said,
“Mathematicians are like Frenchmen. Whatever you say to them,
they translate into their own language, and forthwith it is something
entirely different.” It is not clear whether he was more disgusted with
Frenchmen or mathematicians.
Geometry and not arithmetic being the basis of Greek mathe-
matics, geometrical construction with a compass and a straight edge
was a necessary precondition for any arithmetic operation to be
acceptable. Even with the idea of a point, the idea of zero was absent.
Negative numbers were not imagined either, as a line of negative
length could not be drawn. Rational numbers posed no difﬁculty
as these could be easily obtained by geometrically drawing equal
divisions of a straight line of integer length. Even taking the positive
square root of a positive number was an acceptable mathematical
operation. For a given value of x, the value of
√
x can be constructed
as described below [24, 26].
Referring to Figure 1.1, let the length of the line AB be unity which
deﬁnes the scale of the diagram. The line AB is extended up to the
point C so that BC represents the given number x in the scale of the
drawing. A semicircle is drawn with AC as its diameter. The line BD
perpendicular to AC intersects the semicircle at D. Then, BD is the
Figure 1.1.

Introduction 7
required value of
√
x. Using similar triangles ABD and BDC (note
that ∠ABD = ∠CBD, ∠ADB = ∠DCB), we can write
BD
AB
=
BC
BD
or, BD2
= AB · BC
or, BD =
√
BC =
√
x.
Pythagoreans were really shocked and devastated when they
realised that irrational numbers exist. But the construction of irrational
numbers and hence their existence could not be denied.
Pythagoreans, who did not have number symbols, classified
numbers according to the geometrical figures that could be drawn
by regular arrangements of pebbles or dots on the floor or on the
sand. Following this idea, numbers 3, 6, 10, . . . were called triangular
numbers [18] (see Figure 1.2). Similarly 4, 9, 16, . . . were called
square numbers (see Figure 1.3). It is easy to note that the triangular
numbers can be expressed as n(n + 1)/2, with n representing a
natural number. It may be pointed out that this formula includes the
number 1, which indicates a single point that is unable to define any
shape. The square numbers can be written as m2, with m representing
a natural number.
It is interesting to note that mathematical relationships were
derived using Figures 1.2 and 1.3. For example, by drawing an
inclined line on one side of the diagonal (and parallel to it) of a
Figure 1.2.

Figure 1.3.
square number, indicated by solid lines in Figures 1.3(b) and 1.3(c),
one concludes that every square number is a sum of two consecutive
triangular numbers. Likewise, by drawing a vertical and a horizontal
line, indicated by dashed lines again in Figure 1.3(c), it is easy to
obtain (m + 1)2
= m2 + 2m + 1. Similarly, by counting the dots (or
pebbles) along the diagonal and along lines parallel to the diagonal
in Figures 1.3(a)–1.3(c), one can derive the well-known sum of an AP
series as follows:
22
= 1 + 2 + 1
32
= 1 + 2 + 3 + 2 + 1
42
= 1 + 2 + 3 + 4 + 3 + 2 + 1
.
.
.
n2
= 1 + 2 + 3 + · · · + n + · · · + 3 + 2 + 1
or, 1 + 2 + 3 + · · · + (n − 1) =
1
2
(n2
− n) =
(n − 1)n
2
.
At this stage we may point out that, besides the trivial solution 1,
the following set of inﬁnite numbers, 36(n = 8, m = 6), 1225(n = 49,
m = 35), 41616(n = 288, m = 204), and so on, can be considered as
both triangular and square numbers. In 1733, Leonhard Euler posed
the question “For what values of n, are the triangular numbers also

Introduction 9
square numbers”? He also provided the answerc “Triangular num-
bers with n = 1, 8, 49, 288, 1681, 9800, etc., are square numbers
corresponding to m = 1, 6, 35, 204, 1189, 6930, etc., respectively.”
In the incredible, wonderful world of numbers, mathematicians
discover mysterious connections. For example all the values of m,
for which m2 are also triangular numbers, mentioned above, can be
obtained from the following continued fraction
1
6 − 1
6− 1
6− 1
6−···
(1.1)
By truncating this continued fraction at various stages, we get the
so-called rational convergents, which are seen to be
1
6
,
1
6 − 1
6
=
6
35
,
1
6 − 1
6−1
6
=
35
204
, and so on.
Now we can observe that the numerator and the denominator
of each of these rational convergents, viz., 1, 6, 35, 204, give the
required values of m. The same process continues for ever to yield all
the inﬁnite values of m, for which m2 are also triangular numbers [20].
The continued fraction (1.1) was given by the Indian mathemat-
ical genius S. Ramanujan. Kanigel [16] tells us the following story.
In December 1914, Ramanujan was asked by P
. C. Mahalanobis
(founder of the Indian Statistical Institute) to solve a puzzle, that
appeared in a magazine. The puzzle stated that n houses on one side
of a street are numbered sequentially starting from 1. The sum of the
house-numbers on the left of a particular house having the number
m equals that of the houses lying on the right of this particular house.
With the value of n lying between 50 and 500, one has to determine
the values of m and n.
cC. E. Sandifer (2007). The Early Mathematics of Leonhard Euler, The MAA Tercentary
Euler Celebration. Washington D.C.: The Mathematical Association of America.
pp. 102–105.

According to the puzzle, we can write
1 + 2 + 3 + · · · + (m − 1) = (m + 1) + (m + 2) + · · · + n
or,
m(m − 1)
2
=
(n − m)(m + n + 1)
2
or, m2
=
n(n + 1)
2
. (1.2)
Thus we get back the question posed by Euler regarding the
triangular and square numbers. With 50 < n < 500, there is a
unique answer with m = 204 and n = 288 (see above Euler’s list).
See Appendix A for solution of equation (1.2).
Ramanujan while stirring vegetables over the gas fire, rattled
out the continued fraction (1.1) giving all the infinite values of m.
When asked how he had obtained all the values of m, he answered
“Immediately I heard the problem it was clear that the solution should
obviously be a continued fraction; I then thought which continued
fraction? And the answer came to my mind.”
Ramanujan possibly got the first two values, namely 1 and 6,
mentally. This is not difficult; the corresponding values of n are
1 and 8, respectively. The sums of the house-numbers on either side
of the particular house come out as 0 and 15, respectively. Then it is
only the genius Ramanujan, who can make the leap to construct the
continued fraction (1.1) using only the numbers 1 and 6 to give all
the solutions!

Chapter 2
Integers
2.1. Representation of Integers
Two most common systems of representation of integers are decimal
(base 10) and binary (base 2). The first choice probably has been
dictated by the fact that we have 10 fingers. The latter one is used in
digital computers to represent two possible states as on and off. In
the past, some other systems were also used commonly. For example,
Babylonian astronomers used sexagesimal (base 60). The remnants
of this system are still found in our units of time measurements, viz.,
60 seconds in a minute and 60 minutes in an hour. Likewise we
still consider angle measurements in degrees with 360 degrees in
a full circle. Bases of 12 and 20 were also used in some earlier
civilisations.
In the modern decimal positional notation, ten digit symbols
0, 1, 2, 3, . . . , 9 are used. The last nine digits are also used for
indicating the first nine positive integers. Here we discuss non-
negative integers and by putting a negative (−) sign one can include
negative integers. Any integer I can be written as
I =
n

k=0
ak10k
(2.1)
11

with ak’s representing one of the ten digits. The values of k =
0, 1, 2, . . . , respectively, denote the places of units, tens, hundreds
and so on. Thus any number of (n + 1) digits is written as I =
anan−1an−2 . . . a2a1a0 with an = 0.
In the binary system, an integer I is written as
I =
n

k=0
ak2k
, (2.2)
where ak’s are either 0 or 1, with an = 1. Thus I is a string of 0’s
and 1’s.
The value of n in both equations (2.1) and (2.2) depends on the
size of the integer; a larger value of n implies a larger integer.
Now onwards, we will consistently use the decimal (base 10)
system. It is important to note that some results included in this book
are true only for this system of representation; whereas some other
results are independent of the system of representation.
Geometrically, integers are represented by points on a line as
indicated in Figure 2.1. Consecutive integers are equally spaced on
this number line. The line, starting from zero extends to inﬁnity in both
directions, right and left. Positive integers are located on the right and
negative integers on the left of zero.
2.2. Curious Patterns in Numbers
In this section, we present a small collection of somewhat surprising
patterns in numbers generated by one or more arithmetical oper-
ations, like addition, multiplication and exponentiation, on simple
numbers.
H G
–3 –2 –1 0 1 2 3
F E A B C D K
Figure 2.1.

Integers 13
2.2.1. Multiplication
(a) 1 × 1 = 1
11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321
11111 × 11111 = 123454321
.
.
.
111111111 × 111111111 = 12345678987654321,
(b) 11 × 91 = 1001
11 × 9091 = 100001
11 × 909091 = 10000001
.
.
.
11 × 9090909090909091 = 100000000000000001.
2.2.2. Combination of multiplication and addition
(a) 1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
.
.
.
123456789 × 8 + 9 = 987654321,
(b) 0 × 9 + 1 = 1
1 × 9 + 2 = 11

12 × 9 + 3 = 111
123 × 9 + 4 = 1111
.
.
.
12345678 × 9 + 9 = 111111111,
(c) 0 × 9 + 8 = 8
9 × 9 + 7 = 88
98 × 9 + 6 = 888
987 × 9 + 5 = 8888
.
.
.
98765432 × 9 + 0 = 888888888,
(d) (8 × 8) + 13 = 77
(8 × 88) + 13 = 717
(8 × 888) + 13 = 7117
.
.
.
(8 × 88888888) + 13 = 711111117.
2.2.3. Consecutive integers
(a) 1 + 2 = 3
4 + 5 + 6 = 7 + 8
9 + 10 + 11 + 12 = 13 + 14 + 15
16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24
.
.
.
And the pattern continues.

Integers 15
(b)
32
+ 42
= 52
(Also see Section 2.7)
102
+ 112
+ 122
= 132
+ 142
212
+ 222
+ 232
+ 242
= 252
+ 262
+ 272
362
+ 372
+ 382
+ 392
+ 402
= 412
+ 422
+ 432
+ 442
.
.
.
For the kth line of this pattern, the starting term is (2k2 + k)
2
with
(k + 1) terms on the left and k terms on the right of the equality
sign and the pattern continues forever [31].
(c) It has been shown that the sum of the squares of 24 con-
secutive numbers is a perfect square if the starting number is
1, 9, 20, 25, . . . [28]. For example, we have
12
+ 22
+ 32
+ · · · + 232
+ 242
= 702
92
+ 102
+ 112
+ · · · + 312
+ 322
= 1062
202
+ 212
+ 222
+ · · · + 422
+ 432
= 1582
.
.
.
In fact, all the numbers (12 + 22), (12 + 22 + 32), (12 + 22 +
32 + 42), . . . , (12 + 22 + 32 + · · · + 222 + 232) are non-squares.
This result was known from 1875 when Lucas (see his name
also in Section 2.4.2) said that a square pyramid of cannonballs
contains a square number of balls only when it has 24 balls along
each side of the base. Just imagine this pyramid as 24 layers
of cannonballs with one at the top layer and 4 on the second
topmost layer and so on.
(d) We have
33
+ 43
+ 53
= 63
This is an exception; there is no other such pattern with cubes of
consecutive integers.

2.2.4. Pascal’s triangle
Blaise Pascal, a famous French genius, proposed a pattern of
natural numbers in the form of a triangle which has found many
mathematical applications. Historians tell that the same pattern was
found in a Sanskrit text ‘Chandas Shastra’ written by Pingala.d
The original text, written sometime between 500 and 200 BC, has
not survived. But other Indian mathematicians of the tenth century
mention this. This pattern, known as Pascal’s triangle [47] in the West,
is shown in Figure 2.2. It may be mentioned that this pattern is known
by different names in different countries. In Iran it is called Khayyam
triangle, in China Yang Hui’s triangle and in Italy Tartaglia’s triangle.
Pascal made this triangle popular by applying it in probability theory.
Only six rows are written in Figure 2.2. A little observation of the
entries in this pattern can help the reader to find a rule (more than one
possible) to write the numbers in this pattern as long as one wishes.
Start with 1 at the top. In each subsequent row (to form an ultimate
equilateral triangular array), the number of entries is equal to the
row number. For each entry at a location, go up one row and add
the numbers that you find on immediate left and right. If no number
is noticed, then take it as 0.
It is easy to verify that by adding the numbers in each row one gets
20( = 1), 21, 22, 23, and so on, i.e., 2k−1 for the kth row. If we insert
alternately − and + signs between the entries in a row (obviously,
starting from the second row), then the net result in each row is zero.
Due to symmetry this may be obvious for the even-numbered rows,
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Figure 2.2.
dI. Stewart (2013). Seventeen Equations that Changed the World. London: Profile
Books.

Integers 17
but no such symmetry applies for the odd-numbered rows. We will see
an application of using alternate signs between the entries in a row
of the Pascal’s triangle in Section 3.7.5 for determining the Bernoulli
numbers.
There are many other applications of this versatile pattern. We
mention only a few of these here. Let us count the row number n
starting with 0 denoting the top row, and column number k in each
row starting again with 0 at the left end. Then the kth entry in the
nth row gives the value of nCk, also written as (n
k ), which gives the
number of possible combinations of n things taken k (k n) at a
time. The entries in the nth row also provide the coefﬁcients of the
binomial expansion of (1 + x)n
.
Let us now shift each row of the Pascal’s triangle horizontally so
that the ﬁrst entry in each row appears one below the other as shown
in Figure 2.3.
Now begin from the starting 1’s in each row and add all the
numbers that are encountered if we move diagonally up to the
right. This generates the sequence (1, 1, 2, 3, 5, 8, 13, . . . ), which is
famously known as Fibonacci sequence discussed in Section 2.6.1.
The second numbers in each row of the Pascal’s triangle (Fig-
ure 2.2) are the natural numbers 1, 2, 3, 4, . . . . These numbers
can also be seen in the second column of Figure 2.3. The third
numbers in each row of the Pascal’s triangle (Figure 2.2) are the
so-called triangular numbers (see also Figure 1.2) 1, 3, 6, 10, . . . .
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Figure 2.3.

These numbers, as expected, are also seen in the third column of
Figure 2.3.
2.3. Iterations
Starting from an initial value, the process of carrying out an identical
mathematical operation on the outcome of every stage is called
iteration. Such iterations may end up in a fixed integer or may
exhibit a cyclic (periodic) behaviour producing repeatedly the same
set of integers. The type of end behaviour depends both on the
mathematical operation and the initial value [35]. The ultimate
behaviour is called an attractor and the set of initial values yielding
the same attractor constitute the basin of attraction of that particular
attractor. In this section, we consider some iterations that give rise to
a fixed point and/or a periodic attractor.
2.3.1. Number of even, odd and total digits
Let us start from any arbitrary integer. Then count the number of even
(remember 0 is an even digit), odd and the total number of digits.
Write these three numbers side by side in this order to generate the
new number at the end of the first step of iteration. If we continue the
iteration, then we arrive at a fixed point attractor 123 independent of
the initial value. Thus, for this iteration process, all integers constitute
the basin of attraction of this unique fixed point attractor 123.
For example, let us start with the number 75,816,430,923,
481,061,853,257. This number has 11 even digits, 12 odd digits
and 23 total digits. So the first step of iteration generates the number
111223. Continuing the iteration we get 246 → 303 → 123 →
123 → · · · .
The end result is independent of the initial value, you may try
some other starting integer.
2.3.2. Sum of squares of the digits
Define an iteration process, starting from a positive integer, to
generate a new number by taking the sum of the squares of digits of

Integers 19
the initial number. Then the same process is repeated with the new
number so generated. It has been proved that this iteration process
can generate only two types of attractors depending on the initial
value. One attractor is a fixed point 1, which is obtained from a few
initial values such as 86. With this initial value the iteration continues
as 86 → 100(=82 + 62) → 1(=12 + 02 + 02) → 1 → · · · . The other
attractor is a periodic 8-cycle attractor. With most of the initial values
the iteration reaches one of the numbers in the following 8-cycle
periodic attractor:
89 → 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89 → · · ·
and then comes back to the same number after eight iterations. Which
number in this cycle is reached first depends on the initial value.
For example, if the initial value is 25, then the iteration continues
as 25 → 29(= 22 + 52) → 85(= 22 + 92) → 89(= 82 + 52). After
reaching 89, the same number is repeated after eight iterations as
shown above. If we start with an initial value 4, then the iteration
continues as 4 → 16(= 42) → 37(= 12 + 62). After reaching 37, the
same number is reached again after eight iterations as shown above.
The basin of attraction of the periodic attractor is found to be much
bigger than that of the fixed point 1.
2.3.3. Sum of cubes of the digits
Start with a number divisible by 3. With this restriction on the initial
choice, the iteration is defined as follows. Take the sum of the cubes
of the digits to create the next number. If this iteration is continued,
we arrive at a fixed point attractor 153. For example, let us start with
93. The iteration proceeds as
93 → 756(= 93
+ 33
) → 684 → 792 → 1080
→ 513 → 153 → 153 · · · .
All multiples of 3 constitute the basin of attraction of this fixed
point attractor 153.
It may be mentioned that there exist only three other numbers
(not counting trivial 1) which are sums of the cubes of their own

digits. These numbers are 370, 371 and 407 and unlike 153, none
of these is a multiple of 3.
2.3.4. A fixed point at 1089
Choose any three-digit number with different digits at the places of
units and hundreds. For example, we take 725. Then reverse the
order of the digits to obtain a second number 527. Take the positive
difference of these two numbers, which is 725−527 = 198. Reverse
the order of the digits of this positive difference to get 891. Obtain
the sum of these two numbers as 198+891 = 1089. One can easily
verify and prove that the number 1089 is independent of the initial
choice (725), of course, satisfying the constraint mentioned in the first
sentence of this section.
2.3.5. Kaprekar numbers
First, we consider a three-digit number with at least two different
digits, i.e., one can take 100 but not 333. Then write the largest and
the smallest numbers using these digits. Take the positive difference
to get a new number which completes the first step of iteration. Then
continue the iteration process with the outcome of every stage. We will
observe 495 as the fixed point attractor of this iteration. For example,
let us start with an initial choice 212 (note all digits are not the same
as required) and the process yields
221 − 122 = 099 → 990 − 099 = 891 → 981 − 189
= 792 → 972 − 279 = 693 → 963 − 369
= 594 → 954 − 459 = 495 → 954 − 459
= 495 → · · · .
Exactly the same process if continued with a four-digit (not all
same digits) starting number, one arrives at fixed point attractor 6174.
We may illustrate this with a starting number 1576 as given below:
1576 → 7651 − 1567 = 6084 → 8640 − 0468
= 8172 → 8721 − 1278

Integers 21
= 7443 → 7443 − 3447
= 3996 → 9963 − 3699
= 6264 → 6642 − 2466
= 4176 →→ 7641 − 1467
= 6174 → 7641 − 1467 = 6174 → · · · .
These two numbers 495 and 6174 are known as Kaprekar
numbers, named after D. R. Kaprekar, an Indian recreational number
theorist, who died in 1986.
2.3.6. Collatz conjecture (hailstone numbers)
Let an iteration be deﬁned as follows:
xk+1 =



3xk + 1 if xk is odd,
xk
2
if xk is even.
In 1937, Collatz conjectured that for all initial values the iteration
settles at a periodic attractor . . . , 4, 2, 1, 4, 2, 1, . . .. The conjecture is
yet to be proved. But computer simulation has found no exception
for all initial values up to 5.764 × 1018 [14]. As an example, let us
show the iteration with a starting value 17 as
17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16
→ 8 → 4 → 2 → 1 → 4 → 2 → 1 → . . . .
If at any stage we encounter an odd number, then the next two
numbers are greater than this odd number. This is easy to see as
3xk + 1 is even and greater than xk with odd values of xk. The
next number in the sequence, i.e., 3xk+1
2 , is also greater than xk.
If this number is also even, then the next number in the sequence
3xk+1
4 is less than xk for all xk 1. With an even number at any
stage, the next number is always smaller until one encounters an odd
number. Thus the iteration produces numbers drifting up and down.
The numbers in the sequence are called “hailstone numbers” [28]. In
a storm cloud hailstones drift up and down in a seemingly haphazard

fashion before hitting the ground and then rise a little. The number 1
is the ground level (behaving elastically to raise the sequence again
up to 4). For some initial values, the number of iterations required
to hit the periodic attractor may be very large. For example with a
starting value 27, it requires 111 iterations to reach 1 for the first time
and the highest hailstone number reached is 9,232. Pickover [28]
reports that with starting values 1 to 1000, the highest hailstone
number is 9,232 for more than 350 initial values. With starting
numbers up to 108, the largest number of steps to hit 1 is 949 and
that occurs when the starting number is 63,728,127 [46]. You can
increase the number of iterations indefinitely by considering multiples
of even numbers of the form 2p, with ever increasing value of the
integer p.
2.4. Prime Numbers
The most basic classification of natural numbers is in two groups,
namely, prime and composite numbers. Primes are those numbers
which have only trivial factors 1 and the number itself. The first few
prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, . . . . All prime numbers
except 2 are odd. Prime numbers have continued to draw the atten-
tion of mathematicians for more than two millennia. A prehistoric
bone (dating back to 6500 BC), now famously known as “Ishango
bone” [27, 32], contains four sets of notch marks in a column. The
numbers in these sets are 11, 13, 17, 19, i.e., only the prime numbers
in the range 10–20. Around 1000 BC the special characteristic of
prime numbers was appreciated in Chinese culture. They considered
even numbers as female, odd umbers as male and prime numbers
as macho numbers which resisted all attempts to break them into
smaller factors.
Prime numbers continue to play important roles in modern day
civilisation. The difficulty of factorising a product of two very large
prime numbers is one of the key features of secret information
exchange over the internet [39]. Mathematicians are still busy trying
to establish a pattern in the appearance of prime numbers [32, 10].
The titles of references [32, 10] suggest that the mathematicians are

Integers 23
still obsessed with their attempt to listen to the music of the primes.
Besides this famous problem, there exist other unresolved problems
involving prime numbers [42]. Great interest in prime numbers is
not restricted to only serious mathematicians; it is also widespread in
the vast area of recreational mathematics. A large number of both
types of mathematicians agreed to upload, in a website, their curious
observations about prime numbers. A collection of these results until
2009 is available in [7].
The fundamental theorem of arithmetic states that every integer
can be factorised in a unique way (but for the order) in terms of its
prime divisors. For example, we can write 18 = 2×3×3 (or 3×2×3
and so on). This apparently self-evident theorem requires some subtle
reasoning for its mathematical proof [9].
Prime numbers are classified using different criteria and are
normally named after famous mathematicians. For example, some
prime numbers are generated by formula, such as Euclidean,
Fermat, Mersenne, Pierpoint primes. Another set is defined by some
mathematical criteria; these include Pillai, Ramanujan and Wilson
primes. Yet another group is identified by some pattern, and twin
primes belong to this group. Primes belonging to famous sequences
(see Section 2.6) are named after such sequences, e.g., Fibonacci
primes [47]. Primes of curious characteristics are given some catchy
names as will be seen shortly.
Prime numbers continue forever, i.e., there is no largest prime.
Until today, about 200 different proofs of this statement are
available [2]. Different types of primes, mentioned above, may
be finite or infinite in number. The sets of infinite natural num-
bers and infinite prime numbers are related by a formula given
by Euler, which, famously known as the Golden Key [10, 20], is
given by

p
1
(1 − 1
ps )
=

n
1
ns
s 1, (2.3)
where the left-hand-side products are carried over all the prime
numbers p and the right-hand-side sum is carried over all the natural
numbers n.

2.4.1. Euclidean primes
Euclid provided an elegant proof of infinitude of primes by using the
principle of contradiction. If a largest prime exists, then let it be nth
prime pn. Consider a number P which is 1 more than the product of
all primes up to pn. Thus,
P = 2 × 3 × 5 × 7 × · · · × pn + 1. (2.4)
Obviously P is not divisible by any of the primes up to pn. So either
P itself is a prime greater than pn or has a prime factor greater than
pn. In both cases pn cannot be the largest prime.
Euclidean primes PE are defined as those prime numbers which
are generated by equation (2.4) with different values of n. Note that
not every value of n generates a prime number. So the first few
Euclidean primes are as follows:
with n = 1, PE = 2 + 1 = 3,
with n = 2, PE = 2.3 + 1 = 7,
with n = 3, PE = 2.3.5 + 1 = 31,
with n = 4, PE = 2.3.5.7 + 1 = 211,
with n = 5, PE = 2.3.5.7.11 + 1 = 2311.
With n = 6, equation (2.4) does not yield a prime as
P = 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30,331 = 59 × 509.
The next Euclidean prime (after n = 5) is obtained with pn = 31,
when PE = 200,560,490,131. Then there is a huge gap, as the next
Euclidean prime occurs with pn = 379, when the Euclidean prime
is too large to write here. A very large PE with pn = 24,209 has
been discovered. No one knows whether Euclidean primes continue
forever or there exists a largest Euclidean prime.
2.4.2. Mersenne primes
Numbers of the form Mn = 2n − 1, with n as a prime number, are
named after the French monk and mathematician Marin Mersenne.

Integers 25
Mersenne (1644) claimed Mn is also a prime number for the following
prime values of n:
n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.
Only the prime values of Mn are called Mersenne primes. It is
easy to prove that Mn cannot be a prime unless n is a prime. In
Mersenne’s era, it was not possible to test the primality of a very
large number. Consequently, it is not surprising that later on it has
been found Mn is composite for n = 67 and 257 (both included in the
Mersenne’s list) and Mn is prime for n = 61, 89 and 107 (all missing
in the Mersenne’s list) [43]. The first few Mersenne primes are easily
obtained as M2 = 3, M3 = 7, M5 = 31, M7 = 127, . . . .
Lucas proposed an efficient test for checking the primality
of Mersenne’s numbers 230 years after Mersenne’s primes were
defined. This test was later improved by Lehmer in 1930. According
to this Lucas–Lehmer test, one needs to first define a sequence
L2 = 4, Lk+1 = L2
k − 2 for k ≥ 2, (2.5)
and Mn is a prime if and only if Mn divides Ln for n 2 [32]. We
can easily verify that M3 = 7 divides L3 = 14; M5 = 31 divides
L5 = 37, 634 and so on.
Mersenne primes are a happy hunting ground for very large
prime numbers. In August 2008, a distributed computing project on
the internet called GIMPS (Great Internet Mersenne Prime Search)
obtained a Mersenne prime with n = 43,112,609. This number
has 12,978,189 digits and the mathematics department of UCLA
received $100,000 for crossing 10 million digits for a Mersenne
prime. A larger Mersenne prime with n = 57,885,161 held the record
of the largest Mersenne prime until 2015. This largest (till that date)
Mersenne prime having 17,425,170 digits was the 48th Mersenne
prime known until that time. This record was broken in 2016 when
a Mersenne prime with n = 74,207,281 and having 22,338,618
digits was reported. Incidentally, this is also the largest known prime
number until today. It is not known whether the number of Mersenne
primes is finite or infinite.

Mersenne numbers that are composite have an interesting prop-
erty. All prime factors of composite Mn’s leave a remainder 1 when
divided by 2n [20]. For example, M11 = 2407 = 23 × 89, and
both 23 and 89 leave a remainder 1 when divided by 2 × 11 = 22.
Similarly, M23 = 8388607 = 47×178481, where again both prime
factors leave a remainder 1 when divided by 2 × 23 = 46.
2.4.3. Double Mersenne primes
Double Mersenne numbers, defined in a manner similar to Mersenne
numbers, are given by
MMn
= 22n−1
− 1. (2.6)
It has already been mentioned in Section 2.4.2 that for double
Mersenne number to be a prime, it is necessary (not sufficient) that
(2n − 1) must be a prime, i.e., a Mersenne prime. Thus, double
Mersenne primes must necessarily be of the form (2Mn − 1). The first
four double Mersenne primes are
MM2
= 23
− 1 = 7 = M3,
MM3
= 27
− 1 = 127 = M7,
MM5
= M31,
MM7
= M127.
M127 is a 39-digit number. It has been verified that corresponding
to the next four Mersenne primes, i.e., with n = 13, 17, 19 and 31,
the double Mersenne numbers are composite and their prime factors
have also been explicitly obtained. For the next Mersenne prime, with
n = 61, the corresponding double Mersenne number is too large
(greater than 10694127911065419641
) and present day computers are
not capable of testing its primality. It is conjectured that there exists
no fifth double Mersenne prime [47].
2.4.4. Fermat primes
For a long time mathematicians have tried in vain to produce a
formula to generate if not all primes, at least only primes. One such

Integers 27
Table 2.1.
p Fp
0 3
1 5
2 17
3 257
4 65,537
5 4,294,967,297
effort was by one of the greatest mathematicians, Pierre de Fermat
(1601–1665). He proposed the following formula for generating only
primes
Fp = 22p
+ 1, (2.7)
where p represents non-negative integers 0, 1, 2, 3, . . . . Table 2.1
shows the ﬁrst six values of Fp.
During Fermat’s time it was not possible to verify whether F5 is a
prime or not. In 1732 Euler showed that F5 = 641 × 6,700417 and
hence not a prime. As of 2014, one knows that Fp’s are not primes
for 5 ≤ p ≤ 32. The smallest Fermat number not known to be prime
or composite is F33. But it has been proved that Fp for p = 23,741
is composite with a prime factor 5 × 223,743 [31]. Later it has been
shown that F3329780 is also a composite number with a prime factor
193 × 23329782 + 1.
Fermat primes are associated with possible geometric (using only
compass and straight edge) construction of regular polygons with
prime number of sides. Greek geometers knew how to construct
equilateral triangles (F0 = 3) and regular pentagons (F1 = 5).
The next regular polygon with prime number of sides that can
be geometrically constructed must have 17(=F2) sides. No such
construction is possible for regular polygons having 7, 11 or 13
sides (i.e., the prime numbers between F1 and F2). Gauss proved this
special property of Fermat primes and constructed a regular 17-sided
polygon. Later on construction was carried out for a polygon with

257 sides and it was kept in a large wooden box at the University of
Göttingen.
Gauss was justifiably proud of this discovery he had made at
a very young age. To honour the sentiment Gauss attached to
this achievement, it was decided that the pedestal for his statue at
Braunschweig (his birthplace) will be a 17-sided polygon. The maker
of the monument realised that with so many sides, the pedestal would
appear as a circle and the significance of the number 17 would be
lost. Finally a 17-pointed star, rather than a polygon, was constructed.
2.4.5. Pierpoint primes
A lot of mathematics has been done on the primes named after the
mathematician Pierpoint [47]. Pierpoint primes are prime numbers
of the form
Pu,v = 2u
3v
+ 1, (2.8)
where u and v are non-negative integers. First few of these primes
are
P0,0 = 2, P1,0 = 3, P2,0 = 5, P1,1 = 7, P2,1 = 13,
P4,0 = 17, P1,2 = 18, P2,2 = 37, P3,2 = 73, . . . .
The smallest prime that is not a Pierpoint prime is 11. Up to 2011,
the largest Pierpoint prime that had been reported was P7033641,1 =
3 × 27033641 + 1. It is conjectured that Pierpoint primes continue
forever. The number of Pierpoint primes up to a large N is believed
to be of the order O( log N).
2.4.6. Sophie Germain primes
Sophie Germain (1776–1831), one of the great women mathemati-
cians, contributed significantly towards the solution of some special
cases of the famous Fermat’s Last Theorem (Conjecture). In the
process she considered a special set of primes, which are now called
Sophie Germain primes.
Sophie Germain primes, S, are those primes for which 2S+1 are
also prime numbers. The latter set are called “safe primes”. The first

Integers 29
few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, . . . .
In public key cryptography very large Sophie Germain primes
like 1,846,389,521,368 + 11600 are used. Such primes are also
useful in primality testing. A very large Sophie Germain prime,
18,543,637,900,515 × 2666667 −1, was reported in 2012 [47]. This
number has 200701 digits. The question whether Sophie Germain
primes continue forever or not is still unanswered.
The story of the struggle of Sophie Germain to learn and do math-
ematics must be retold at every opportunity. Contemporary French
society was against any woman taking up mathematics seriously.
Initially even her parents created a lot of hurdles to dissuade her
from doing mathematics. When they relented, Sophie Germain took
Lagrange’s mathematics courses by presenting false identification
as Monsieur Le Blanc. She used the same pseudonym in her initial
correspondences with Gauss. Gauss was highly impressed with her
work and expressed the highest level of admiration particularly when
her real identity was revealed. He even convinced the University of
Göettingen to award her an honorary degree. Unfortunately, Sophie
Germain died before the formalities could be completed.
2.4.7. Pillai primes
A prime number p for which there is a positive integer n such that n!
is 1 less than a multiple of p, but p is not 1 more than a multiple of
n is called a Pillai prime. It may be recalled at this stage that mod( · )
represents the one-way modulo function. This function is defined as
b = a mod (n) implying that b divided by n leaves a remainder a. It
should be noted that for given values of n and b, we can determine
a uniquely. But with given values of a and n, b cannot be determined
uniquely. One can only write b = kn + a, with k as any integer.
Thus, mathematically a Pillai prime Pp is written, using the modulo
function, as
n! = −1 mod (Pp), but Pp = 1 mod (n) (2.9)
These primes are named after Indian mathematician S. S.
Pillai (1901–1950). The first few Pillai primes are 23, 29, 59, 61,

67, 71, . . . . For Pp = 23, one can verify n = 14, when n! =
14! = 87,178,291,200. It has been proved that Pillai primes continue
forever. Pillai was considered only next to Ramanujan amongst the
Indian mathematicians who went to Cambridge in the twentieth
century.
2.4.8. Ramanujan primes
The nth Ramanujan prime Rn is the smallest integer for which
π(x) − π
x
2

≥ n for all x ≥ Rn, (2.10)
where π(x) is the prime counting function giving the number of primes
up to and including x.
These primes are named after the Indian mathematical genius
S. Ramanujan (1887–1920) who reported these primes while giving
an alternate proof of Bertrand’s conjecture. This conjecture claiming
for every n 1, there exists at least one prime p for n p 2n
was earlier proved by Russian mathematician Chebyshev. The first
few Ramanujan primes are
R1 = 2, R2 = 11, R3 = 17, R4 = 29, R5 = 41, . . . .
Note that Rn is necessarily a prime and π(Rn) − π(Rn
2 ) = n.
Obviously, Ramanujan primes continue forever.
2.4.9. Wilson primes
Wilson conjectured that for every prime p, (p − 1)! + 1 is divisible
by p. Lagrange proved this and also its converse, namely that
whenever (p−1)!+1 is divisible by p, p is a prime. Wilson primes are
defined as those prime numbers Wp for which W2
p divides (Wp−1)!+1.
The only known Wilson primes are 5, 13 and 563. It has been verified
that there is no other Wilson prime at least up to 500,000,000.
2.4.10. Twin primes
Except 2, all prime numbers are odd. If two consecutive odd numbers
are both prime numbers, then these constitute a pair of twin primes.

Integers 31
For example, (3, 5), (5, 7), (11, 13), (17, 19), . . . , (1019, 1021), . . .
are twin primes. In 1919, Vigo Brun proved that the sum of the
reciprocals of twin primes converges. The series
Htp =

1
3
+
1
5
+

1
5
+
1
7
+

1
11
+
1
13
+ · · · +

1
1019
+
1
1021
+ · · · (2.11)
converges to a value, which is known as Brun’s constant. Incidentally,
it may be mentioned that the sum of the reciprocals of the infinitely
many primes diverges very slowly. In mathematical terms, this sum
diverges as O( log log N), which is so slow that a joke is that “yes it
diverges, but no one has seen it diverge”! The numerical value of the
Brun’s constant is 1.9021605824 . . . .
In 1994, Robert Nicely obtained the above-mentioned value of
the Brun’s constant. During this computation, Nicely observed what
is now infamously known as the Intel Pentium division bug. He found
while obtaining the reciprocals of twin primes 824,633,702,441
and 824,633,702,443 that erroneous results are obtained by the
computer. This confirmed that there is a bug in the FPU (floating
point unit) of Pentium processors. In 1995 Intel announced a pre-tax
charge of $475 million for replacement of the processors. This has
been identified as the highest amount of money associated with any
mathematical activity [13].
As we go along the integers, twin primes become incredibly
sparse. In 1986, a very large twin prime pair having 2259 digits
was identified as 107,570,463 × 102250
± 1. In 2012, twin prime
pair having 200,700 digits was obtained as 3, 756, 801, 695, 685×
2666,699 ± 1; until 2015, this was the largest known twin prime
pair. It is still not known whether twin primes continue forever or
not. Had the series Htp, given by equation (3.8), not converged,
one could have easily concluded that the twin primes continue
forever. In 2013, it was proved that there exist infinite prime-pairs
which differ by a number less than 70 million. This has to be
brought down to 2 for proving the “twin prime conjecture” which

states that twin primes continue forever. The number 70 million
was considered a good starting point, and has since been brought
down to 246.
2.4.11. Carmichael numbers
Carmichael numbers are not prime numbers. In fact, it is known that
these odd numbers must have at least three prime factors. The reason
for including the story of Carmichael numbers in this section is that
these numbers satisfy a mathematical relationship that is also satisfied
by all prime numbers. Carmichael numbers are also referred to as
pseudoprimes.
From Fermat’s Little Theorem, it is known that, for all primes p,
the following statement is true. If any integer b is not divisible by p
(i.e., b and p are co-primes; in other words have no common divisor
except trivial 1), then bp−1 when divided by p leaves a remainder 1.
Mathematically one writes, for co-primes b and p
bp−1
= 1 mod (p). (2.12)
For example, for p = 7 with b = 2, one gets 26 = 64 = 1 mod (7).
Besides all prime numbers, there exists a set of infinite composite
numbers, called Carmichael numbers, C, co-prime to b, which also
satisfy equation (2.12). One can thus write, for co-primes b and C
bC−1
= 1 mod (C). (2.13)
The lowest Carmichael number is 561. One can verify that
2560
= 1 mod (561). (2.14)
The next two Carmichael numbers are 1105 and 1729. The latter
one is also known as “Taxicab number” (see Section 2.8). A large
Carmichael number having 1,101,518 prime factors has also been
reported. This number has more than 16 million digits. It has been
proved that for a large value of N, there are at least N2/7 Carmichael
numbers ≤ N. Such prevalence of Carmichael numbers prevents the
use of equation (2.12) for primality testing. This equation can only
provide a necessary but not sufficient condition for a prime number.

Integers 33
It is known that Carmichael numbers (C) are square-free (i.e., no
prime factor is repeated) odd composite numbers for which (C − 1)
is divisible by (pi − 1), where pi’s represent all the prime factors of
C. We can easily verify this statement for the two lowest Carmichael
numbers given above. For example, 561( = 3 × 11 × 17) − 1 = 560
is divisible by 2( = 3 − 1), 10( = 11 − 1) and 16( = 17 − 1). Similarly
1105( = 5 × 13 × 17) − 1 = 1104 is divisible by 4( = 5 − 1), 12( =
13 − 1) and 16( = 17 − 1).
2.4.12. “emirp”
This word has been created by writing “prime” backward. If the
digits of a prime number when written in the reverse order create a
different prime, then such a number is called emirp. There are many
emirp’s, like 13 (the smallest emirp) and 389 since 31 and 983 are
also primes. 1,597 is another example of an emirp. This particular
number is famous in the context of Brahmagupta’s equation (also
commonly known as Pell’s equation).
Brahmagupta’s equation,
x2
− Dy2
= 1, (2.15)
seeks integer solutions of x and y where D is a non-square positive
integer. With D a square number, equation (2.15) implying two
square numbers differing by 1 has only trivial solution x = 1 and
y = 0. For non-square values of D, a set of inﬁnite solutions can
be easily obtained after one obtains the smallest solutions (see
Appendix A). For D = 1,597, the smallest solutions become astro-
nomically large [5], x is a 48-digit number and y is a 47-digit number!
2.4.13. Cyclic primes
Starting from an n-digit prime number, if the ﬁrst digit is brought to the
end to generate a new prime number and the process is continued
with the number so generated and all the numbers turn out to be
prime until the starting number is reached after n steps, then such
numbers are called n-digit cyclic primes. Examples of 4-digit and

6-digit primes are:
1193 → 1931 → 9311 → 3119 → 1193 → . . .
193939 → 939391 → 393919 → 939193
→ 391939 → 919393 → 193939 → · · · .
2.4.14. Prime digit/composite digit primes
All the digits in such a prime represent prime numbers (like 2, 3, 5, 7).
Example of prime digit primes are 23,2357. Similarly, if all the digits
of a prime number represent non-prime numbers, then it is called a
composite digit prime; for example, 64486949 is a composite digit
prime.
2.4.15. Almost-all-even-digits primes
Except 2, all other even numbers are composites. So a prime number
must end with an odd digit. All primes which have only an odd digit
at the end with all other digits even (0 is an even digit) are called
almost-all-even-digits primes. If the number has more than one digit,
then such a prime is also called a single-odd-digit prime. A typical
example is 86420864207.
2.4.16. Palindromic and plateau primes
A palindrome is a word which reads the same forward and backward,
e.g., “noon”, “radar”, etc. Exactly the same way a palindromic prime
is defined as a prime number which remains the same when its
digits are written in the reversed order. An example is the prime
number 133020331. The following four palindromic primes are in
an arithmetic sequence with a common difference 810:
13931, 14741, 15551 and 16361.
A subgroup of palindromic primes is defined as plateau primes
when the same internal repeated digit is confined between smaller
digits at the two ends. For example, 355555553 and 1777771 are
plateau primes.

Integers 35
2.4.17. Snowball primes
A prime number which remains a prime at every stage as we write it
from the left is called a snowball prime. For example, 73939133
is a snowball prime, since 7, 73, 739, 7393, 73939, 739391,
7393913 also are all primes. A snowball remains a snowball as
it grows in size while rolling on a snow-covered ground. Here the
sequence represents various stages of the snowball (analogue of
prime). Mathematically such primes are also called right-truncatable
primes. Staring from the original number we can continue to truncate
one digit at a time from the right end and still get prime numbers at
every stage.
2.4.18. Russian Doll primes
In the famous Russian Matryoshka nested doll, as one removes
the outer most layer, a similar smaller doll appears at every stage.
Following this analogy, a Russian Doll prime is deﬁned as a prime
number which continues to remain a prime number as remove
one digit at a time from the left end. For example, 4632647 is a
Russian Doll prime, since 632647, 32647, 2647, 647, 47 and 7
are all prime numbers. Mathematically such primes are called left-
truncatable primes. Another common example of such a prime is
33333331.
2.4.19. Pandigital primes
A prime using all the ten digits 0 to 9 at least once is called a pandig-
ital prime. One example of many such primes is 10123456789.
2.4.20. Very large prime numbers with
repeated pattern
Large prime numbers cannot be easily memorised. Writing a large
prime number from memory becomes an easy task if there is a simple
repetitive pattern of digits. For example, we can easily memorise the
following 28-digit prime number
1234567891234567891234567891,

where the sequence 1–9 is repeated three times followed by 1.
Following the notation used in [7], we can write the above number
as (123456789)31, where the subscript 3 indicates that the number
within the parenthesis is repeated side by side three times. Using this
notation, an easy-to-remember 841-digit prime can be written as
(10987654321234567890)421.
This number is called an almost equipandigital prime as all nine
digits, except 1, are used same number of (84) times and the digit 1
is used 85 times.
A still larger patterned prime with 3793 digits can be written as
(1676)9481. In this number odd (1 and 7) and even (6) digits appear
alternately and consequently this number is called an alternate digit
prime.
2.4.21. Miscellany
Mathematics with prime numbers is an old but a still living subject.
There are still old and new unproven conjectures, which are very
simple to state. The most famous of these is the “Goldbach conjec-
ture”. In 1742, Goldbach wrote in a letter to Euler that “Any even
number greater than 2 (remember 2 is the only even prime number)
can be expressed as a sum of two primes (repetitions allowed).”
This almost 250 years old conjecture is yet to be proved. But no
counterexample has been found when checked for all even numbers
up to 1018. The best proven result, so far as this conjecture is
concerned, is that every large even number is a sum of a prime and
a “semiprime” [9]. A semiprime is a product of at most two primes. In
2000, a mathematical ﬁction Uncle Petros Goldbach’s Conjecture
was written by Apostolos Doxiadis. The publisher announced a
million-dollar prize for a proof of the conjecture within two years.
The prize remained unclaimed, but this enjoyable book had a good
advertisement.
Another conjecture made in 1985 by Andrica is also yet to
be proved. Andrica’s conjecture states “The difference between the

Integers 37
square roots of two consecutive prime numbers is less than 1.”
Mathematically, one writes this conjecture as
√
pn+1 −
√
pn 1,
where pn is the nth prime number. This conjecture has been verified
up to a large value of n of the order of 1016 [27]. It is easy to show
that this conjecture implies the gap between the nth prime and the
next is less than 2
√
pn +1. So far the maximum value of
√
pn+1 −
√
pn
is
√
11 −
√
7 ≈ 0.67087.
One very important problem in prime numbers is to test whether
a given number is prime or composite, and in the latter case what
are its prime factors. Even Gauss considered these two as important
problems in arithmetic. The brute force method to answer these two
questions becomes very computer-time consuming for large num-
bers. If the given number N has n digits, then the computation time
increases roughly as 10n/2. This exponential growth is undesirable
and such a problem is called an NP (non-polynomial) problem. For
computation, people would like to have a P-type problem where the
computational effort grows at a fixed power of n. Until 2002, all P-type
algorithms for primality testing were probabilistic, implying we can
get a wrong answer (though rarely). In 2002, three Indians Agarwal–
Kayal–Saxena (AKS) for the first time showed that a P-class algorithm
exists for testing of primality [42]. In this proposed algorithm, the
computational time grows at a rate not faster than n12. The exponent
has now been brought down to 6. But these P-type algorithms are still
not competitive with the probabilistic algorithms for the ranges of N
which are currently under consideration. In future, if the exponent of
n can be brought down to around 3 or less, then this deterministic
algorithm will be as useful as the probabilistic ones. Fortunately, no
such algorithm is available for the prime factorisation problem, which
ensures the security of secret information exchange over the internet.
For some discussions on two famous theorems involving prime
numbers, viz., “the Prime Number Theorem” (PNT) and Fermat–Euler
Theorem on two types of primes, see Sections 3.8.6 and 4.6,
respectively.

2.5. Composite Numbers
According to Fundamental Theorem of Arithmetic, mentioned in
Section 2.4, any composite number C can be written as
C = p
n1
1 × p
n2
2 × p
n3
3 × · · · × p
ni
i , (2.16)
where pi’s are prime numbers, p1 = 2, p2 = 3, p3 = 5 and so on
with ni’s as non-negative integers (including 0).
2.5.1. Highly composite numbers
Indian mathematical genius S. Ramanujan failed twice to clear
his college examinations in Madras (present day Chennai). He
finished his 3-hour mathematics paper in 30 minutes but did very
poorly in all other subjects, flunked in physiology. When he went
to Cambridge, he wanted to graduate. Professor Hardy arranged
to waive the requirement of examinations and course work as
Ramanujan liked to do mathematics alone in a room. Hardy wanted
him to get a BA degree by research, and asked him to submit his
52 page (printed) paper on highly composite numbers that was
published in the Proceedings of the London Mathematical Society
in late 1915. The paper was so long, that it had its own contents
page. Hardy found the work highly original, though away from
the main channel of mathematical research. The proofs of the
assertions were elementary but highly ingenious. Based on this paper
Ramanujan received the BA degree from Cambridge University in
1916 [16].
Prime numbers have only two trivial divisors, namely 1 and
the number itself. A composite number has other divisors besides
these two trivial ones. A highly composite number is defined as a
number which has more distinct divisors than all composite numbers
less than it [14]. The first highly composite number is 6 which
has two divisors, viz., 2 and 3. The only composite number less
than 6 is 4, which has only one divisor, i.e., 2. The following list
shows first few highly composite numbers (n) and their number
of divisors d(n). The number itself and 1 are also counted as a
divisor.

Integers 39
Highly composite numbers (n) Number of divisors d(n)
6 4
12 6
24 8
36 9
Ramanujan listed all the highly composite numbers up
to 6,746,328,386,800. He overlooked only one number, viz.,
29,331,862,500 in this long list.
He also claimed that for a highly composite number, the expo-
nents of equation (2.16) satisfy the following relation
n1 ≥ n2 ≥ n3 ≥ n4 ≥ · · · .
For example, we can observe the following highly composite numbers
6 = 21
× 31
12 = 22
× 31
24 = 23
× 31
.
.
.
332, 640 = 25
× 33
× 51
× 71
× 111
43, 243, 200 = 26
× 33
× 52
× 71
× 111
× 131
2, 248, 776, 129, 600 = 26
× 33
× 52
× 72
× 111
× 131
×171
× 191
× 231
.
Ramanujan also showed that the last exponent must neces-
sarily be 1. An impressive asymptotic formula for the number of
highly composite numbers up to a very large number N was also
found.
2.5.2. Sierpinski’s numbers
In Section 2.4.4, we have seen that Fermat failed in producing
a formula which generates only prime numbers. Another great
mathematician Euler tried with the following two formulae to generate

only prime numbers:
n2 − n + 41, (2.17)
n2 − 79n + 1601 (2.18)
with n representing all non-negative integers. Equation (2.17) gen-
erates 40 primes with n = 0, 1, 2, 3, . . . , 40 (the first two values of n
generate the same prime number 41). This formula fails at n = 41,
as one gets a square number 41×41. Equation (2.18) generates 80
prime numbers with n = 0, 1, 2, 3, . . . , 79. This formula also fails at
n = 80 generating the same square number 41×41. A question can
be asked how about writing a formula for generating only composite
numbers. Except 2, all even numbers are composite, so generating
even composite numbers is trivial. How about generating only odd
composite numbers by a formula?
Sierpinski showed that there are infinitely many odd numbers S
for which the formula S × 2n + 1 generates only (obviously odd)
composite numbers for all values of the natural number n. These
special values of S are called Sierpinski’s numbers. It is believed
that the smallest Sierpinski’s number is 78,557. Only six numbers
(viz., 10223, 21181, 22699, 24737, 55459) need to be checked for
confirming this belief, but obviously the check is not easy as it involves
all n. The composite numbers 78,557 × 2n +1 must have one of the
following prime factors:
(3, 5, 7, 13, 19, 37, 73).
The next known Sierpinski’s number is 271729 for which the
covering set of factors is (3, 5, 7, 13, 17, 241) [47].
2.5.3. Perfect and associated numbers
Consider a composite number and all its proper divisors, including 1
but excluding the number itself. If the sum of the proper divisors of a
number equals the number itself, then the number is called a perfect
number. The first few perfect numbers are 6, 28, 496 and we may
verify that
(i) proper divisors of 6 are 1, 2 and 3 and also 6 = 1 + 2 + 3;

Integers 41
(ii) proper divisors of 28 are 1, 2, 4, 7, 14 and also 28 = 1 + 2 +
4 + 7 + 14;
(iii) proper divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and
also 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.
In terms of iterations defined as taking the sum of proper divisors
to produce the next number, one can say that a perfect number
depicts fixed point behaviour and also constitutes its unique basin
of attraction.
Euler proved Euclid’s observations that all even perfect numbers
are of the form 2n−1 × (2n − 1) where (2n − 1) is a prime number
(called Mersenne prime Mn — see Section 2.4.2). As already noted in
Section 2.4.2, only 49 Mersenne primes and therefore only 49 even
perfect numbers are known so far (until 2016). From Euler’s formula,
these even perfect numbers are seen to be of the form Mn
i=1 i, where
Mn is a Mersenne prime. From Section 2.4.2, we see that the first
three Mersenne primes are M2 = 3, M3 = 7, M5 = 31. Thus the
first three perfect numbers according to this sum are 1 + 2 + 3 =
6, 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 and 1 + 2 + 3 + · · · + 31 = 496
as mentioned earlier. Pickover [28] writes about a sensational but
false claim of a 155 digit perfect number that was reported in a
newspaper in 1936. It may be pointed out that every even perfect
number, except 6, can be written as a sum of cubes of consecutive
odd integers starting from 1. For example,
28 = 13
+ 33
,
496 = 13
+ 33
+ 53
+ 73
,
8,128 = 13
+ 33
+ 53
+ 73
+ 93
+ 113
+ 133
+ 153
.
Till today no odd perfect number has been found, but their non-
existence has not been proved either.
There are many numbers which are just one more than the sum
of their proper divisors. These numbers are called slightly excessive
numbers [40]. It is easy to see that the proper divisors of 2n are
1, 2, 22, 23, . . . , 2n−1 with n as any natural number. Thus the sum of
the proper divisors (which form a geometric progression) is 2n − 1,

i.e., one less than the number. Consequently, all numbers of the
form 2n are slightly excessive numbers. If a number is one less than
the sum of its proper divisors, then it is called a slightly deficient
number. Till today no such slightly deficient number has been found,
but their non-existence has not been proved. It may be mentioned
that there is no dearth of numbers which are generally deficient (also
known as ‘abundant’ numbers), i.e., less than the sum of their proper
divisors, (though not by just one), e.g., 12 with proper divisors (1, 2, 3,
4, 6) which add up to 16. Even two and three consecutive deficient
numbers have been located [28]. Two consecutive numbers 5,775
and 5,776 are deficient. One can verify that the proper divisors
of 5, 775 (= 3 × 52 × 7 × 11) add up to 6,129 and that of
5, 776 (= 24 × 192
) add up to 6,035. The smallest odd deficient
number is 945. Every integer greater than 20161 may be written as
the sum of two deficient numbers. The same is also true for all even
integers greater than 46 [30].
A multiply perfect number is defined as one for which the sum
of proper divisors is an integral multiple of the number. There are
many examples of such numbers. For example, the proper divisors
of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40 and 60,
which add up to 240. The proper divisors of 672( = 25 × 3 × 7) add
up to 1344.
2.5.4. Friendly (Amicable) numbers
Two numbers are called ‘Friendly numbers’ or ‘Amicable numbers’
if the sum of proper divisors of one gives the other number and
vice versa. One can say friendly numbers display a two-cycle
periodic behaviour with those two numbers constituting the basin
of attraction. Greek mathematicians discovered 220 and 284 are
friendly numbers. We can easily verify that
(i) the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55,
110 and their sum is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 +
55 + 110 = 284 and
(ii) the proper divisors of 284 are 1, 2, 4, 71, 142 and their sum is
1 + 2 + 4 + 71 + 142 = 220.

Integers 43
Pythagoreans defined friendship through these two numbers.
They said a friend is “one who is the other I such as 220 and 284”.
In 1636 Fermat gave the second set of Friendly numbers as
17,296 and 18,416. Soon after, Descartes gave another set as
9,363,584 and 9,437,056. Euler listed 60 such pairs. But all these
great mathematicians missed the pair 1,184 and 1,210 pointed out
by 16-year old Paganini in 1866 [40].
2.5.5. Sociable numbers
Mathematicians have discovered “sociable numbers” of period 5
(5-cycle) when five numbers form a loop, where the sum of the proper
divisors of one number gives the next number in the loop. After five
steps the original starting number is retrieved. One example is
12,946 → 14,288 → 15,742 → 14,536
→ 14,264 → 12,946 → · · · .
Sociable numbers depicting a period of 28 have also been
noted [21, 23]. This cycle of 28 numbers can be written as
14,316 → 19,116 → · · · → 629,072 → · · ·
→ 19,916 → 17,716 → 14,316 → · · · .
2.5.6. Untouchable numbers
Paul Erdös defined most unfriendly or untouchable numbers as those
which cannot be the sum of the proper divisors of any number. The
first few untouchable numbers are 2, 5, 52, 88, 96, 120, . . . . We may
note that in the above list only 2 and 5 are prime numbers. It is
believed that except these two primes all other untouchable numbers
are even and hence composite numbers. That justifies inclusion of
untouchable numbers in this section on composite numbers.
2.5.7. Smith numbers
A Smith number is a composite number, the sum of whose digits
equals that of the digits of its prime divisors. Pickover [28] recounts

a brief history of these curious numbers named after the brother-
in-law of the mathematician who proposed this number in 1982.
The telephone number of Mr. Smith, 493-7775 was the highest such
number known at that time. One may verify that 4,937,775 = 3 ×
5 × 5 × 65837 and the sum of the digits of this number is 4 + 9 +
3 + 7 + 7 + 7 + 5 = 42. The sum of the digits of its prime divisors
is 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42. Two consecutive Smith
numbers 728 and 729 are called Smith brothers. A formula has been
discovered for generating very large Smith numbers and one with
2,592,699 digits has been identiﬁed. A palindromic Smith number is
12,345,554,321.
2.6. Sequences
Besides the commonly known arithmetic sequence and geometric
sequence there are some other famous sequences of natural numbers
which have generated a lot of mathematical activity. These sequences
are generated following some rules, which may or may not have
originated from modelling of any physical process. In this chapter,
we discuss a few such sequences [37].
2.6.1. Fibonacci (Hemachandra) sequence
In 1202, Leonardo of Pisa, better known as Fibonacci, introduced a
sequence in his book Liber Abaci. In the western world, this sequence
is named after him as Fibonacci sequence. He considered a highly
hypothetical growth model for rabbit population. In this model,
all rabbits are immortal. However, the sequence so produced has
such interesting mathematical properties and connections to other
branches of mathematics that mathematicians are still producing
new results [10]. In fact a journal Fibonacci Quarterly, established in
1963, is devoted entirely to mathematics related to Fibonacci
sequence. Later on in Sections 3.8.2 and 3.8.3 we will discuss
applications of this sequence in various ﬁelds of study.
Fibonacci sequence is written as
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . .

Integers 45
Table 2.2.
No. of rabbit couples
Month Baby Adult Total
1 1 0 1
2 0 1 1
3 1 1 2
4 1 2 3
5 2 3 5
6 3 5 8
This sequence of natural numbers is generated with the nth term
Fn = Fn−1 + Fn−2 for n ≥ 3 (2.19)
with F1 = 1 and F2 = 1.
Fibonacci considered the following model for the growth of rabbit
population. Start with a rabbit couple, which becomes adult after
1 month. Every adult couple continues to breed another couple at
an interval of 1 month. With this growth rule, the number of rabbit
couples during each subsequent month will be as shown in Table 2.2.
During any month, the total number of rabbit couples is the
number in the previous month plus that in the last but one month;
each one of the latter category producing a baby couple. It is easy to
note the Fibonacci numbers in the total numbers of rabbit couples,
mentioned in the last column of Table 2.2. The same numbers also
appear in the numbers of baby and adults rabbit couples from the
third and second month, respectively.
The fatal ﬂaw of the immortality of the rabbit couples, implied in
the population dynamics model described above, can be eliminated
by considering a little variation in the model as discussed now. Again
we start with a rabbit couple and assume that each couple generates
a couple of rabbits in each of the successive two months and then
they die. Now the number of rabbit couples “born” in the nth month
gives the Fibonacci number Fn satisfying again equation (2.19) with
F1 = 1 and F2 = 1.

July 26, 2017 11:35 The Story of Numbers - 9in x 6in b2870-ch02 page 46
Table 2.3.
Total number
No. of total beats Types of rhythms of rhythms
1 S 1
2 SS, L 2
3 SSS, LS, SL 3
4 SSSS, LSS, SLS, SSL, LL 5
5 SSSSS, LSSS, SLSS, SSLS,
LLS, SSSL, LSL, SLL 8
It must be mentioned that more than 50 years before Fibonacci,
Hemachandra (in 1150) arrived at the same sequence of natural
numbers from an altogether different consideration. Hemachandra
was a great Sanskrit scholar, poet and linguist. He considered
different possible rhythms of total n beats consisting of short (S —
one beat) and long (L — two beats) syllables. The total num-
bers of rhythms for various numbers of total beats are listed in
Table 2.3.
In a particular stage one adds one beat (S) to the previous stage
and two beats (L) to the last but one stage to generate all possible
rhythms. Note that the total number of rhythms are 1, 2, 3, 5, 8, … ,
which can be generated by
Hn = Hn−1 + Hn−2 for n ≥ 3 with H1 = 1 and H2 = 2. (2.20)
Another way of arriving at Hemachandra sequence is to consider
the total number of different ways of expressing various integers,
n (total number of rhythms) as sums of 1 (equivalent to S) and 2
(equivalent to L). Hemachandra described this method of generating
his numbers (equation (2.20)) in a one-line shloka. For example, for
H5 we write
5 = 1 + 1 + 1 + 1 + 1 (which is equivalent SSSSS)
5 = 2 + 1 + 1 + 1, 5 = 1 + 2 + 1 + 1 + 1 and so on.

Integers 47
Now onwards, no distinction will be made between Fibonacci and
Hemachandra sequences by noting
Fn+1 = Hn for n ≥ 1 with F1 = 1.
It has been proved that in these sequences 144 is the only non-
trivial (disregarding trivial 1) perfect square and 8 is the only non-
trivial perfect cube. No other number in these sequences is an integral
power of any natural number. The first few primes in these sequences
are 2, 3, 5, 13, 89, 233, 1597 (see Section 2.4.12), 28,657 and
514,229 and so on. The largest known prime in the sequence has
more than thousands of digits. It is not known whether the number
of primes in these sequences is finite or infinite [43].
Prime-free Fibonacci-like sequences generated by equa-
tion (2.19) can be easily obtained by choosing F1 and F2 as two
composite numbers having a common divisor, like 10 and 15 or
simply 2 and 4. Graham was the first to show that by taking the
following two co-prime (no common divisor) numbers as F1 and F2,
one can generate a prime-free Fibonacci-type sequence [14]:
F1 = 5,794,765,361,567,513 and
F2 = 20,615,674,205,555,510.
2.6.2. Padovan sequence
Professor Ian Stewart [44] named a Fibonacci-like sequence the
Padovan sequence. The numbers in this sequence are generated by
the formula
Pa(n + 1) = Pa(n − 1) + Pa(n − 2) for n ≥ 2 with
Pa(0) = Pa(1) = Pa(2) = 1. (2.21)
Padovan numbers Pa(n) for different values of n are listed in Table 2.4.
It may be noted that Padovan numbers also satisfy
Pa(n + 1) = Pa(n) + Pa(n − 4) for n ≥ 4. (2.22)
For example, we can see, from Table 2.4,
Pa(10) = Pa(9) + Pa(5) and Pa(16) = Pa(15) + Pa(11).

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