(Dolciani Mathematical Expositions)
Mathematical Association of America | 2009 | 152 páginas | pdf | 660 kb
A Guide to Elementary Number Theory is a 140 pages exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams.
Though Plato did not quite say, He is unworthy of the name of man who does not know which integers are the sums of two squares he came close. This Guide can make everyone more worthy.
Contents
Introduction . . . . vii
1 Greatest Common Divisors . . . 1
2 Unique Factorization . . . . 7
3 Linear Diophantine Equations . . 11
4 Congruences . . . 13
5 Linear Congruences . . 17
6 The Chinese Remainder Theorem.. . 21
7 Fermat’s Theorem.. . 25
8 Wilson’s Theorem .. . 27
9 The Number of Divisors of an Integer . . 29
10 The Sum of the Divisors of an Integer . . 31
11 Amicable Numbers. . 33
12 Perfect Numbers . . 35
13 Euler’s Theorem and Function . . 37
14 Primitive Roots and Orders . . 41
15 Decimals . . 49
16 Quadratic Congruences . 51
17 Gauss’s Lemma . . 57
18 The Quadratic Reciprocity Theorem . . 61
19 The Jacobi Symbol . . 67
20 Pythagorean Triangles . . 71
21 x4 + y4 ¤ z4 . . . . 75
22 Sums of Two Squares . . . . 79
23 Sums of Three Squares . . 83
24 Sums of Four Squares .. . 85
25 Waring’s Problem . . 89
26 Pell’s Equation . . . 91
27 Continued Fractions . .. . 95
28 Multigrades . .. . 101
29 Carmichael Numbers . . . . . 103
30 Sophie Germain Primes. . . 105
31 The Group of Multiplicative Functions . . . 107
32 Bounds for .pi(x) . . 111
33 The Sum of the Reciprocals of the Primes .. 117
34 The Riemann Hypothesis . . 121
35 The Prime Number Theorem . . 123
36 The abc Conjecture . 125
37 Factorization and Testing for Primes . 127
38 Algebraic and Transcendental Numbers . 131
39 Unsolved Problems . . 135
Index . .. . . 137
About the Author . . . . 141
Contents
Introduction . . . . vii
1 Greatest Common Divisors . . . 1
2 Unique Factorization . . . . 7
3 Linear Diophantine Equations . . 11
4 Congruences . . . 13
5 Linear Congruences . . 17
6 The Chinese Remainder Theorem.. . 21
7 Fermat’s Theorem.. . 25
8 Wilson’s Theorem .. . 27
9 The Number of Divisors of an Integer . . 29
10 The Sum of the Divisors of an Integer . . 31
11 Amicable Numbers. . 33
12 Perfect Numbers . . 35
13 Euler’s Theorem and Function . . 37
14 Primitive Roots and Orders . . 41
15 Decimals . . 49
16 Quadratic Congruences . 51
17 Gauss’s Lemma . . 57
18 The Quadratic Reciprocity Theorem . . 61
19 The Jacobi Symbol . . 67
20 Pythagorean Triangles . . 71
21 x4 + y4 ¤ z4 . . . . 75
22 Sums of Two Squares . . . . 79
23 Sums of Three Squares . . 83
24 Sums of Four Squares .. . 85
25 Waring’s Problem . . 89
26 Pell’s Equation . . . 91
27 Continued Fractions . .. . 95
28 Multigrades . .. . 101
29 Carmichael Numbers . . . . . 103
30 Sophie Germain Primes. . . 105
31 The Group of Multiplicative Functions . . . 107
32 Bounds for .pi(x) . . 111
33 The Sum of the Reciprocals of the Primes .. 117
34 The Riemann Hypothesis . . 121
35 The Prime Number Theorem . . 123
36 The abc Conjecture . 125
37 Factorization and Testing for Primes . 127
38 Algebraic and Transcendental Numbers . 131
39 Unsolved Problems . . 135
Index . .. . . 137
About the Author . . . . 141
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