Mathematical Marvels First Steps in Number Theory A Primer on Divisibility by Shailesh Shirali Universities Press RMO INMO Mathematics Olympiad IMO

169 Pages·2019·47.18 MB·English

by  Shailesh A Shirali

>) Universities Press —— MATHEMATICS.
Mathematical Marvels

FIRST STEPS IN
NUMBER THEORY

A Primer on
DIVISIBILITY

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Shailesh Shirali
Mathematical Marvels

FIRST STEPS IN
NUMBER THEORY

A Primer on

DIVISIBILITY

Shailesh Shirali

Universities Press
Contents

Preface
1 Introduction

1.1 What Is Number Theory?
1.2 Terms And Symbols,

2 Congruences

2.1 Introduction
2.2 Odd And Even Numbers
2.3 The Remainder Symbol

2.4 Other Divisors

2.5 The Congruence Symbol

2.6 Using Congruences

2.7 A Test For Divisibility By 13
2.8 Using Negative Remainders
3 The Elementary Cases

3.1 Introduction

3.2 Divisibility By 2

3.3 Divisibility By 5

34 Divisibility By 4

3.5 A Generalization

4 The Remaining Cases

4.1 Introduction
4.2 Divisibility By 9

ibility By 11

; General Results
4.5 Divisibility By 101

4.6 Divisibility By 7 And 13
4,7 Divisibility By 27 And 37
48 Other Divisors

4.9 Concluding Remarks

5 A Different Approach

5.1 Introduction
4 FIRST STEPS IN NUMBER THEORY: A PRIMER ON DIVISIBILITY

5.2 Divisibility By 7
533 Another Formulation

54 Searching For New Rules
5.5 Divisibility By 17

5.6 Divisibility By 53

5.7 Divisibility By 11

6 Non-Decimal Bases

6.1 Introduction
6.2 Divisibility By 2

6.3 When d Divides b— 1
6.4-When d Divides b+ 1
6.5 The Remaining Cases
6.6 Concluding Remarks

7 Special Topics

7.1 Appetizers

7.2 GCDs

73 The Two Jug Problem
7.4 Wilson's Theorem

75 Fermat's “Little” Theorem
7.6 The Divisor Function
1.7 The Factorial Numbers
7.8 Two Applications

7.9 Pythagorean Triples
7.10 Pell’s Equation

7.11 Automorphic Numbers
7.12 Consecutive Integers
7.13 Sums of Reciprocals
7.14 Primality Testing

7.15 Polynomials

8 Miscellaneous Problems

8.1 Problems
8.2 Solutions
8.3 The Last Word

Further Reading
Mathematical Olympiads

Appendix C: Solutions
Index
Preface

You have no doubt encountered the following statement:

A number is divisible by 9 if and only if the sum of its digits is
divisible by 9.

This statement gives a test for divisibility by 9. In this book,
you will read about other such tests and about the rich theory
behind them. En route, you will encounter a subject called Number
Theory. To study this book, all you really need is familiarity with
elementary arithmetic and algebra (addition and subtraction of
algebraic expressions, the laws of exponents, the idea of prime
factorization of an integer, the notion of relative primeness of two
integers, etc); in short, material which would normally be covered
in classes 7-9 in most countries. It is amazing how far one can
go from these simple beginnings!

You must be familiar with the tests of divisibility by divisors such
as 2, 3, 4, 5, 6, 8, 9, 10 and 11. Of these, the test for divisibility
by 2 was probably the one you first came across, followed by the
one for divisibility by 10, then the one for divisibility by 5, ...;
and last of all, the one for divisibility by 11. Perhaps it struck you
as strange that there seemed to be no tests for divisibility by 7
and 13, The logic behind the tests may also have been a source
of puzzlement; for instance, the test for divisibility by 11. In this
book, we shall study such tests. We shall also devise tests for
divisibility by numbers such as 7, 13, 17, 19, 23, ..., and explain
why the tests for divisibility by 2, 3, 4, 5, ..., 11 are so simple,
whereas those for divisibility by 7, 13, 17, 19, ..., are relatively
more complicated.

Except for Chapter 6, we shall be working throughout in the
base-10 system (with place values based on the powers of 10; so
when we say ‘453’, what we refer to is the number (4 x 100) +(5 x
10) +(3x1)). There are other number systems— the binary system,
based on the powers of 2, the ‘enary system, based on powers
of 3, and so on. In Chapter 6, we explore tests of divisibility for
numbers expressed in other bases.
‘vil FIRST STEPS IN NUMBER THEORY: A PRIMER ON DIVISIBILITY

Chapters 7 and 8 are more advanced than the earlier chapters
and meant to be studied by those with a much deeper interest
in the subject. However, they are quite self-contained and can be
understood without having to refer to other textbooks.

The chapter contents are briefly described below.

[) sntroduetion
This chapter is, as its title indicates, an introductory one. The
first section imparts a sense of what number theory is about.
Brief bits of history are included, and also definitions of the
unfamiliar terms used in the book.

Congruences
‘This introduces an extremely important and useful concept in
elementary number theory—that of congruences.

LB) the Elementary Cases
Here, we study the tests for divsiblity by the numbers 2, 4, 8,
16, ..., 5, 25, 125, ..., and by the products of these numbers.

[4] The Remaining Cases :
In this chapter, we study tests for divisibility by odd numbers
which are not powers of 5; that is, by numbers such as 3, 9,
11, 13, 17, 19, ...

[5] 4 Different Approach
‘This chapter introduces a very different approach to the tests
of divisiblity; it is more iterative in nature.

[G] tests OF Divisibitity In Other Bases
Here, we study extensions of the familiar tests to the cases
where the numbers are expressed in non-decimal bases.

Special Topics -
In this chapter, several “bread-and-butter” topics of eleme:
tary number theory are studied, though not in a textbook-ish
PREFACE ix

manner. A lot of interesting material is presented and several
problems are discussed, along with some curious applications.

Miscellaneous Problems
This contains just what the title suggests— problems, problems,
and more problems; all woven in some way around the theme
of divisibility. Many of the problems have their origins in the
Olympiads.

Exercises will be found in plenty, scattered through the book.
These are meant to be done! As has wisely been said, mathematics
is not a spectator sport—one learns and begins to appreciate the
subject only after starting to “do” it. This may be true of any
subject, but nowhere more so than in the case of Mathematics.
At the end of the book solutions are given in full. These should
be consulted only when all else fails!

Acknowledgements

The author gratefully acknowledges the help received from Dr
‘A Kumaraswamy and Mr V Sundararaman, with regard to the
computer software (I4TX) and the PCs used in typesetting this
book; the feedback provided by Professor Phoolan Prasad, Dr
Jayant Kirtane and Shri P K Srinivasan on the content and style
of the book; and the support provided by his wife Padmapriya
and by his colleagues and friends Dr Radhika Herzberger and
Professor Hans Herzberger. He also thanks Universities Press for
the support he has received from them.

Dedication

1 dedicate this book to my parents Shri Ashok R Shirali and
‘Smt Lata A Shirali.
Chapter 1

Introduction

1.1 What Is Number Theory?

The study of divisibility tests offers an excellent introduction to
the subject called Number Theory, also sometimes known as the
“higher arithmetic”. We start this chapter by giving you an idea of
what this subject is all about. Listed below are some topics that
belong to number theory. Alongside each topic are listed some
typical problems in that particular topic.

‘The prime number sequence A prime number p is one whose
only positive divisors are 1 and p. Here is the sequence of prime
numbers: 2, 3, 5, 7, 11, 13, 17, 19, .... Many facts are known
about the primes, for instance,

te.

‘« The number of primes is inf
‘« If nis a positive integer such that 2" ~1 is prime, then n is
prime.
Example 2” — 1 = 127 is prime, and so is 7.
‘« Every prime that is 1 more than a multiple of 4 can be written

as a sum of two squares, and primes that are 1 less than a
multiple of 4 cannot be written in this form.

Example The primes 29 and 73, both of which are 1 more
than some multiple of 4 (29 = 1+4-7, 73 = 144-18), can be
written in this form:

29=57+27, 73 =87 +37.
You can verify for yourself that the primes 11, 19 and 71
cannot be written as sums of two squares.

* This was known to the Greeks.
2 FIRST STEPS IN NUMBER THEORY: A PRIMER ON DIVISIBILITY

Patterns in divisibility Here are some sample results:

# A number is divisible by 3 if and only if the sum of its digits
is divisible by 3.

Example 252 is divisible by 3, and so is 2+5+
257 is not divisible by 3, and neither is 2+5+7= 14.

Let n be any integer, and let N =n? +1. Then each odd
divisor of N is of the form a? +6? for some integers a, b.

9; but

Example Let n= 13; then N = 170, and the odd divisors of
NV are 5, 17 and 85. Observe that 5
and 85 = 9? +22,

‘* Let p be a prime number such that 2° — 1 is not prime. Then
each divisor of 2°—1 is 1 more than a multiple of 2p.

Example 11 is prime, but 2!" ~ 1 = 2047 is not prime. The
proper divisors of 2047 are 23 and 89 (2047 = 23 x 89). Note
that 1, 23, 89 and 2047 are all of the form 1 plus a multiple
of 22.

* Let K be a power of 2 such that 2* +1 is nor a prime number.
‘Then each prime factor of 2 +1 is of the form ak +1 for
some integer a.

Example It is known that 2°? +1 is not prime; this was first
discovered by Euler, who showed that 641 is one of its prime
factors, Observe that 641 = 20-32 +1, that is, 641 is of the
form 32a +1, with a = 20.

Solutions of equation:
ems:
‘« Find all pairs of positive integers m,n such that 2" and 3°
differ by 1 (e.g, 2 and 3? differ by 1).
‘« Find an integer lying between 0 and 1000 that leaves a remain-
der of 1 when divided by 7, a remainder of 2 when divided by
11, and a remainder of 3 when divided by 13,

integers Here are a few sample prob-

37

‘= Can positive integers a,b,c be found such that a® + 5°

Functions defined on the integers Here are two functions that
have been studied very intensively by number theorists:
CHAPTER |, INTRODUCTION. 3

© For z > 0, let f(z) be the number of prime numbers less than
or equal to 2; thus (10) = 4, f(20) = 8, .... This function is
of great interest to number theorists. A typical question: Can
(1000000) be calculated without having to list all the primes
less than 1000000?

«Let p(n) be the number of ways that the positive integer n
can be written as a sum of positive integers. Thus, p(4) = 5,
because 4 can be written as a sum of positive integers in the
following five ways:

4, S41, 242, 24141, 1414141,

whereas p(5) = 7, because 5 can be written as a sum of positive
integers in the following seven ways:

5, 441, 342, 34141, 24241,
Qt1+141, 141414141

This is the partition function. There is a wonderful formula
found by Ramanujan and Hardy that allows us to calculate
the values of p(n) with ease. As n increases, p(n) grows very
rapidly, as the following display shows:

(50) = 204226, p(100) = 190569292,

(200) = 3972999029388,

Number theory has a glorious history behind it. The Greeks were
deeply interested in it, perhaps because of the natural elegance
and simplicity of the subject, but also for philosophic reasons: to
them the universe was “made up” of numbers. They investigated
the relationship between numbers and beauty; for instance, the
appeal of certain shapes in art (e.g., the golden rectangle) and
chords in music. They even constructed the musical scale:

Do—Re—Me-Fa~So—La—Te—Do,
(in the Indian system:

Sa—Re~Ga—Ma—Pa—Da—Ni—Sa)
‘on a numerical basis. (Actually, so did the Indians.) There are
two particularly beautiful theorems of elementary number theory

dating from Greek times that are stated in Euclid’s ancient text,
The Elements: