A Pathway Into Number Theory,9780521575409

A Pathway Into Number Theory

by
Edition: 2nd
ISBN13: 9780521575409
ISBN10: 0521575400
Format: Paperback
Pub. Date: 1996-12-28
Publisher(s): Cambridge University Press
List Price: $108.07

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Summary

Number theory is concerned with the properties of the natural numbers: 1, 2, 3 … During the seventeenth and eighteenth centuries, number theory became established through the work of Fermat, Euler and Gauss. With the hand calculators and computers of today the results of extensive numerical work are instantly available and the road leading to their discoveries may be traversed with comparative care. Now in its second edition, this book consists of a sequence of exercises that will lead readers from quite simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A modern secondary school course in mathematics is sufficient background for the whole book which is designed to be used as an undergraduate course in number theory to be pursued by independent study without supporting lectures.

Table of Contents

Preface to the second edition xi(2)
Introduction xiii
1 The fundamental theorem of arithmetic
1(21)
1-24 Division algorithm
1(6)
25-42 Greatest common divisor and Euclidean algorithm
7(2)
43-61 Unique factorisation into primes
9(4)
62-66 Infinity of Primes
13(1)
67 Mersenne primes
13(1)
Summary
14(1)
Historical note
14(2)
Notes and answers
16(6)
2 Modular addition and Euler's function
22(26)
1-18 Congruence classes and the Chinese remainder theorem
22(5)
19-38 The groups (Zn +) and their generators
27(6)
39-56 Euler's function
33(3)
57-64 Summing Euler's function over divisors
36(1)
Summary
37(1)
Historical note
38(1)
Notes and answers
39(9)
3 Modular multiplication
48(31)
1-20 Fermat's theorem
48(5)
21-25 Wilson's theorem
53(1)
26-33 Linear congruences
53(1)
34-42 Fermat-Euler theorem
54(1)
43-44 Simultaneous linear congruences
55(1)
45-57 Lagrange's theorem for polynomials
56(5)
58-74 Primitive roots
61(3)
75-87 Chevalley's theorem
64(2)
88-95 RSA codes
66(1)
Summary
67(1)
Historical note
68(2)
Notes and answers
70(9)
4 Quadratic residues
79(18)
1-29 Quadratic residues and the Legendre symbol
79(2)
30-43 Gauss' lemma
81(3)
44-65 Law of quadratic reciprocity
84(3)
Summary
87(1)
Historical note
88(1)
Notes and answers
89(8)
5 The equation xn + yn = zn, for n = 2, 3, 4
97(22)
1-18 The equation x2 + y2 = z2
97(3)
19-23 The equation x4 + y4 = z4
100(1)
24-26 The equation x2 + y2 + z2 = t2
101(1)
27-68 The equation x3 + y3 = z3
102(6)
Summary
108(1)
Historical note
108(2)
Notes and answers
110(9)
6 Sums of squares
119(21)
1-36 Sums of two squares
119(4)
37-52 Sums of four squares
123(3)
53-54 Sums of three squares
126(1)
55-61 Triangular numbers
126(1)
Summary
127(2)
Historical note
129(1)
Notes and answers
130(10)
7 Partitions
140(14)
1-15 Ferrers' graphs
140(1)
16-35 Generating functions
141(4)
36-47 Euler's theorem
145(2)
Summary
147(1)
Historical note
147(1)
Notes and answers
148(6)
8 Quadratic forms
154(33)
1-20 Unimodular transformations
154(4)
21-31 Equivalent quadratic forms
158(4)
32-43 Discriminant
162(2)
44-52 Proper representation
164(1)
53-72 Reduced forms
165(3)
73-77 Automorphs of definite quadratic forms
168(1)
Summary
169(1)
Historical note
170(1)
Notes and answers
171(16)
9 Geometry of numbers
187(27)
1-28 Subgroups of a square lattice
187(5)
29-46 Minkowski's theorem in two dimensions
192(5)
47-66 Subgroups of a cubic lattice
197(3)
67-73 Minkowski's theorem in three dimensions
200(1)
74-86 Legendre's theorem on ax2 + by2 + cz2 = 0
201(3)
Summary
204(1)
Historical note
204(2)
Notes and answers
206(8)
10 Continued fractions
214(28)
1-7 Irrational square roots
214(1)
8-25 Convergence
214(6)
26-53 Purely periodic continued fractions
220(3)
54-71 Pell's equation
223(3)
72-77 Lagrange's theorem on quadratic irrationals
226(1)
78-82 Automorphs of the indefinite form ax2 - by2
227(2)
Summary
229(1)
Historical note
230(2)
Notes and answers
232(10)
11 Approximation of irrationals by rationals
242(15)
1-10 Naive approach
242(1)
11-22 Farey sequences
243(2)
23-33 Hurwitz' theorem
245(2)
34-43 Liouville's theorem
247(3)
Summary
250(1)
Historical note
250(1)
Notes and answers
251(6)
Bibliography 257(3)
Index 260

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