Friendly Introduction to Number Theory, A

by Silverman, Joseph H

• ISBN13: 9780130309549
• ISBN10: 0130309540
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Pearson College Div

Summary

For courses in Elementary Number Theory for non-math majors, for mathematics education students, and for Computer Science students. This is an introductory undergraduate text designed to entice non-math majors into learning some mathematics, while teaching them to think mathematically at the same time. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results.

Table of Contents

Preface	v
Introduction	1	(6)
What Is Number Theory?	7	(6)
Pythagorean Triples	13	(6)
Pythagorean Triples and the Unit Circle	19	(4)
Sums of Higher Powers and Fermat's Last Theorem	23	(4)
Divisibility and the Greatest Common Divisor	27	(7)
Linear Equations and the Greatest Common Divisor	34	(9)
Factorization and the Fundamental Theorem of Arithmetic	43	(9)
Congruences	52	(6)
Congruences, Powers, and Fermat's Little Theorem	58	(6)
Congruences, Powers, and Euler's Formula	64	(4)
Euler's Phi Function	68	(7)
Prime Numbers	75	(7)
Counting Primes	82	(6)
Mersenne Primes	88	(4)
Mersenne Primes and Perfect Numbers	92	(10)
Powers Modulo m and Successive Squaring	102	(7)
Computing kth Roots Modulo m	109	(4)
Powers, Roots, and ``Unbreakable'' Codes	113	(7)
Euler's Phi Function and Sums of Divisors	120	(5)
Powers Modulo p and Primitive Roots	125	(10)
Primitive Roots and Indices	135	(7)
Squares Modulo p	142	(8)
Is -- 1 a Square Modulo p? Is 2?	150	(11)
Quadratic Reciprocity	161	(11)
Which Primes Are Sums of Two Squares?	172	(12)
Which Numbers Are Sums of Two Squares?	184	(6)
The Equation X4 + Y4 = Z4	190	(3)
Square-Triangular Numbers Revisited	193	(10)
Pell's Equation	203	(5)
Diophantine Approximation	208	(9)
Diophantine Approximation and Pell's Equation	217	(8)
Primality Testing and Carmichael Numbers	225	(11)
Number Theory and Imaginary Numbers	236	(14)
The Gaussian Integers and Unique Factorization	250	(17)
Irrational Numbers and Transcendental Numbers	267	(16)
Binomial Coefficients and Pascal's Triangle	283	(11)
Fibonacci's Rabbits and Linear Recurrence Sequences	294	(12)
Generating Functions	306	(11)
Sums of Powers	317	(11)
Cubic Curves and Elliptic Curves	328	(13)
Elliptic Curves with Few Rational Points	341	(6)
Points on Elliptic Curves Modulo p	347	(12)
Torsion Collections Modulo p and Bad Primes	359	(4)
Defect Bounds and Modularity Patterns	363	(6)
Elliptic Curves and Fermat's Last Theorem	369	(2)
Further Reading	371	(1)
Appendix A. Factorization of Small Composite Integers	372	(2)
Appendix B. A List of Primes	374	(3)
Index	377