Number Theory: A Very Short Introduction
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THE MONGOLS Morris Rossabi
MOONS David A. Rothery
MORMONISM Richard Lyman Bushman
MOUNTAINS Martin F. Price
MUHAMMAD Jonathan A. C. Brown
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NETWORKS Guido Caldarelli and Michele Catanzaro
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NORTHERN IRELAND Marc Mulholland
NOTHING Frank Close
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NUCLEAR POWER Maxwell Irvine
NUCLEAR WEAPONS Joseph M. Siracusa
NUMBER THEORY Robin Wilson
NUMBERS Peter M. Higgins
NUTRITION David A. Bender
OBJECTIVITY Stephen Gaukroger
OCEANS Dorrik Stow
THE OLD TESTAMENT Michael D. Coogan
THE ORCHESTRA D. Kern Holoman
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ORGANIZATIONS Mary Jo Hatch
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PAIN Rob Boddice
THE PALESTINIAN-ISRAELI CONFLICT Martin Bunton
PANDEMICS Christian W. McMillen
PARTICLE PHYSICS Frank Close
PAUL E. P. Sanders
PEACE Oliver P. Richmond
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PERCEPTION Brian Rogers
THE PERIODIC TABLE Eric R. Scerri
PHILOSOPHY Edward Craig
PHILOSOPHY IN THE ISLAMIC WORLD Peter Adamson
PHILOSOPHY OF BIOLOGY Samir Okasha
PHILOSOPHY OF LAW Raymond Wacks
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POLITICAL PHILOSOPHY David Miller
POLITICS Kenneth Minogue
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PSYCHOLOGY Gillian Butler and Freda McManus
PSYCHOLOGY OF MUSIC Elizabeth Hellmuth Margulis
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PSYCHOTHERAPY Tom Burns and Eva Burns-Lundgren
PUBLIC ADMINISTRATION Stella Z. Theodoulou and Ravi K. Roy
PUBLIC HEALTH Virginia Berridge
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THE QUAKERS Pink Dandelion
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READING Belinda Jack
THE REAGAN REVOLUTION Gil Troy
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THE REFORMATION Peter Marshall
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RENAISSANCE ART Geraldine A. Johnson
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RISK Baruch Fischhoff and John Kadvany
RITUAL Barry Stephenson
RIVERS Nick Middleton
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ROCKS Jan Zalasiewicz
ROMAN BRITAIN Peter Salway
THE ROMAN EMPIRE Christopher Kelly
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ROUSSEAU Robert Wokler
RUSSELL A. C. Grayling
RUSSIAN HISTORY Geoffrey Hosking
RUSSIAN LITERATURE Catriona Kelly
THE RUSSIAN REVOLUTION S. A. Smith
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Robin Wilson
Number Theory
A Very Short Introduction
Great Clarendon Street, Oxford OX2 6DP, United Kingdom
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Contents
List of illustrations
1 What is number theory?
2 Multiplying and dividing
3 Prime-time mathematics
4 Congruences, clocks, and calendars
5 More triangles and squares
6 From cards to cryptography
7 Conjectures and theorems
8 How to win a million dollars
9 Aftermath
Further reading
Index
List of illustrations
1 Euclid; Fermat; Euler; Gauss
Granger Historical Picture Archive/Alamy Stock Photo; Lebrecht Music & Arts/Alamy Stock Photo; The State Hermitage Museum, St. Petersburg. Photo © The State Hermitage Museum/photo by E.N. Nikolaeva; akg-images
2 The integers
3 The first four non-zero squares
4 Right-angled triangles
5 18 is a multiple of 3, and 3 is a factor of 18; b is a multiple of a, and a is a factor of b
6 If d divides a and b, then it also divides
7 Two gears with 90 and 54 teeth
8 A periodical cicada
David C. Marshall/Wikimedia Commons (CC BY-SA 4.0)
9 The division rule
10 Special cases of the division rule
11
12
13
14 The sum of the first few odd numbers is a square
15 If b is odd, then b2 has the form
16 Casting out nines
17 A German postage stamp commemorates Adam Riese; An example from Abraham Lincoln’s ‘Cyphering book’
Deutsche Bundespost; George A. Plimpton Papers, Rare Book & Manuscript Library, Columbia University in the City of New York
18 Factorizations of 108 and 630
19 A postage stamp celebrates the discovery in 2001 of the 39th Mersenne prime
Courtesy of Liechtensteinische Post AG
20 Some regular polygons
21 Constructing an equilateral triangle
22 Doubling the number of sides of a regular polygon
23 A 12-hour clock
24 A 7-day clock
25 Some solutions of the Diophantine equation
26 Bachet’s translation of Diophantus’s Arithmetica
Bodleian Library, University of Oxford (Saville W2, title page)
27 A postage stamp celebrates Andrew Wiles’s proof of Fermat’s last theorem
Courtesy of Czech Post
28 A necklace with five beads
29 Shuffling cards
30 The distribution of primes
31 The graph of the natural logarithm
32 The graphs of π(x) and x/log x
33 (a) Bernhard Riemann
Familienarchiv Thomas Schilling/Wikimedia Commons
(b) Riemann’s 1859 paper
Wikimedia Commons
34 Summing the powers of 1/2
35 Points on the complex plane
36 The zeros of the Riemann zeta function in the complex plane
Chapter 1
What is number theory?
Consider the following questions:
In which years does February have five Sundays?
What is special about the number 4,294,967,297?
How many right-angled triangles with whole-number sides have a side of length 29?
Are any of the numbers 11, 111, 1111, 11111, … perfect squares?
I have some eggs. When arranged in rows of 3 there are 2 left over, in rows of 5 there are 3 left over, and in rows of 7 there are 2 left over. How many eggs have I altogether?
Can one construct a regular polygon with 100 sides if measuring is forbidden?
How many shuffles are needed to restore the order of the cards in a pack with two Jokers?
If I can buy partridges for 3 cents, pigeons for 2 cents, and 2 sparrows for a cent, and if I spend 30 cents on buying 30 birds, how many birds of each kind must I buy?
How do prime numbers help to keep our credit cards secure?
What is the Riemann hypothesis, and how can I earn a million dollars?
As you’ll discover, these are all questions in number theory, the branch of mathematics that’s primarily concerned with our counting numbers, 1, 2, 3, …, and we’ll meet all of these questions again later. Of partic
ular importance to us will be the prime numbers, the ‘building blocks’ of our number system: these are numbers such as 19, 199, and 1999 whose only factors are themselves and 1, unlike 99 which is and 999 which is . Much of this book is concerned with exploring their properties.
Number theory is an old subject, dating back over two millennia to the Ancient Greeks. The Greek word ἀριθμὸς (arithmos) means ‘number’, and for the Pythagoreans of the 6th century bc ‘arithmetic’ originally referred to calculating with whole numbers, and by extension to what we now call number theory—in fact, until fairly recently the subject was sometimes referred to as ‘the higher arithmetic’. Three centuries later, Euclid of Alexandria discussed arithmetic and number theory in Books VII, VIII, and IX of his celebrated work, the Elements, and proved in particular that the list of prime numbers is never-ending. Then, possibly around ad 250, Diophantus, another inhabitant of Alexandria, wrote a classic text called Arithmetica which contained many questions with whole number solutions.
After the Greeks, there was little interest in number theory for over one thousand years until the pioneering insights of the 17th-century French lawyer and mathematician Pierre de Fermat, after whom ‘Fermat’s last theorem’, one of the most celebrated challenges of number theory, is named. Fermat’s work was developed by the 18th-century Swiss polymath Leonhard Euler, who solved several problems that Fermat had been unable to crack, and also by Joseph-Louis Lagrange in Berlin and Adrien-Marie Legendre in Paris. In 1793 the German prodigy Carl Friedrich Gauss constructed by hand a list of all the prime numbers up to three million when he was aged just 15, and shortly afterwards wrote a groundbreaking text entitled Disquisitiones Arithmeticae (Investigations into Arithmetic) whose publication in 1801 revolutionized the subject. Sometimes described as the ‘Prince of Mathematics’, Gauss asserted that
Mathematics is the queen of the sciences, and number theory is the queen of mathematics.
The names of these trailblazers will reappear throughout this book (see Figure 1).
1. From left to right; Euclid, Pierre de Fermat, Leonhard Euler, and Carl Friedrich Gauss.
More recently, the subject’s scope has broadened greatly to include many other topics, several of which feature in this book. In particular, there have been some spectacular developments, such as Andrew Wiles’s proof of Fermat’s last theorem (which had remained unproved for over 350 years) and some exciting new results on the way that prime numbers are distributed.
Number theory has long been thought of as one of the most ‘beautiful’ areas of mathematics, exhibiting great charm and elegance: prime numbers even arise in nature, as we’ll see. It’s also one of the most tantalizing of subjects, in that several of its challenges are so easy to state that anyone can understand them—and yet, despite valiant attempts by many people over hundreds of years, they’ve never been solved. But the subject has also recently become of great practical importance—in the area of cryptography. Indeed, somewhat surprisingly, much secret information, including the security of your credit cards, depends on a result from number theory that dates back to the 18th century.