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Robin WilsonNumber TheoryA 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 PictureArchive/Alamy Stock Photo;Lebrecht Music & Arts/Alamy StockPhoto; The State HermitageMuseum, St. Petersburg. Photo© The State HermitageMuseum/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/WikimediaCommons (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, ColumbiaUniversity 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 LiechtensteinischePost 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 ThomasSchilling/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 1What 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 particular 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
