A friendly introduction to number theory/ Joseph H. Silverman

Includes index

What is Number Theory? -- Pythagorean Triples -- Pythagorean Triplesand the Unit Circle -- Sums of Higher Power and Fermat's Last Theorem -- Divisibility and the Greatest Common Divisor -- Linear Equations and the Greatest Common Divisor -- Factorization and the Fundamental Theorem of Arithmetic -- Congruences -- Congruences, Power, and Euler's Formula -- Euler'sPhi Function and the Chienese Remainder Theorem -- Prime Numbers -- Counting Prime -- Mersenne Primes -- Mersenne Primes and Perfect Numbers -- Powers Modulo m and Successive Squaring -- Computing kth Roots and Modulo m -- Powers, Roots, and "Unbreakable" Codes -- Primality Testing and Carmichael Numbers -- Squares Modulo p -- Quadratic Reciprocity -- Proof of Quadratic Reciprocity -- Which Primes Are Sums of Two Squares? -- Which Numbers are Sums of Two Squares? -- Euler's Phi Function and Sums of Divisors -- Powers Modulo p and Primitive Roots -- Primitive Roots and Indices -- The Equation X^4+Y^4=Z^4 -- Square-Triangular Numbers Revisited -- Pell's Equation -- Diophantine Approximation -- Diophantine Approximation and Pell's Equation -- Number Theory and Imaginary Numbers -- The Gaussian Integers and Unique Factorization -- Irrational Numbers and Transcendental Numbers -- Binomial Coefficients and Pascal's Triangle -- Fibonacci's Rabbits and Linear Recurrence Sequences -- Cubic Curves and Elliptic Curves -- Elliptic Curves with Few Rational Points -- Points on Elliptic Curves Modulo p -- Torsion Collections Modulo p and Bad Primes -- Defect Bounds and Modularity Patterns -- Elliptic Curves and Fermat's Last Theorem -- The Topsy-Turvy World of Continued Fractions -- Continued Fractions and Pell's Equation -- Generating Functions -- Sums of Powers -- Appendix: A List of Primes

An introductory text designed to entice non-math majors to think mathematically. 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.