Elementary Number Theory

by Strayer, James K.

• ISBN13: 9781577662242
• ISBN10: 1577662245
• Edition: Revised
• Format: Hardcover
• Copyright: 2001-12-01
• Publisher: Waveland Pr Inc

Table of Contents

Introduction	1	(2)
Divisibility and Factorization	3	(35)
Divisibility	3	(7)
Prime Numbers	10	(8)
Greatest Common Divisors	18	(4)
The Euclidean Algorithm	22	(4)
The Fundamental Theorem of Arithmetic	26	(10)
Concluding Remarks	36	(2)
Student Projects	36	(2)
Congruences	38	(38)
Congruences	38	(10)
Linear Congruences in One Variable	48	(6)
The Chinese Remainder Theorem	54	(5)
Wilson's Theorem	59	(4)
Fermat's Little Theorem; Pseudoprime Numbers	63	(5)
Euler's Theorem	68	(5)
Concluding Remarks	73	(3)
Student Projects	73	(3)
Arithmetic Functions	76	(27)
Arithmetic Functions; Multiplicativity	76	(5)
The Euler Phi-Function	81	(5)
The Number of Positive Divisors Function	86	(3)
The Sum of Positive Divisors Function	89	(3)
Perfect Numbers	92	(4)
The Mobius Inversion Formula	96	(5)
Concluding Remarks	101	(2)
Student Projects	101	(2)
Quadratic Residues	103	(25)
Quadratic Residues	103	(5)
The Legendre Symbol	108	(9)
The Law of Quadratic Reciprocity	117	(9)
Concluding Remarks	126	(2)
Student Projects	126	(2)
Primitive Roots	128	(26)
The Order of an Integer; Primitive Roots	128	(6)
Primitive Roots for Prime Numbers	134	(6)
The Primitive Root Theorem	140	(6)
Index Arithmetic; nth Power Residues	146	(6)
Concluding Remarks	152	(2)
Student Projects	152	(2)
Diophantine Equations	154	(27)
Linear Diophantine Equations	154	(5)
Nonlinear Diophantine Equations; a Congruence Method	159	(2)
Pythagorean Triples	161	(3)
Fermat's Last Theorem	164	(4)
Representation of an Integer as a Sum of Squares	168	(11)
Concluding Remarks	179	(2)
Student Projects	180	(1)
Continued Fractions	181	(50)
Rational and Irrational Numbers	181	(9)
Finite Continued Fractions	190	(5)
Convergents	195	(7)
Infinite Continued Fractions	202	(12)
Eventually Periodic Continued Fractions	214	(9)
Periodic Continued Fractions	223	(5)
Concluding Remarks	228	(3)
Student Projects	229	(2)
A Few Applications	231	(20)
A Recreational Application	231	(2)
Cryptography; The RSA Encryption System	233	(6)
Primality Testing	239	(2)
Pell's Equation	241	(7)
Concluding Remarks	248	(3)
Student Projects	249	(2)
Appendices	251	(21)
A Mathematical Induction	251	(6)
B Equivalence Relations	257	(4)
C Abstract Algebra	261	(3)
D The Binomial Theorem	264	(2)
E Tables	266	(6)
Hints and Answers to Selected Exercises	272	(12)
Bibliography	284	(3)
Index	287