約瑟夫 H.西爾弗曼（Joseph H. Silverman） 擁有哈佛大學(xué)博士學(xué)位。他目前為布朗大學(xué)數(shù)學(xué)教授，之前曾任教于麻省理工學(xué)院和波士頓大學(xué)。1998年，他獲得了美國數(shù)學(xué)會Steele獎的著述獎，獲獎著作為《The Arithmetic of Elliptic Curves》和《Advanced Topics in the Arithmetic of Elliptic Curves》。 他的研究興趣是數(shù)論、橢圓曲線和密碼學(xué)等。
Contents Introduction......................................................... 1 1 What Is Number Theory?............................................. 6 2 Pythagorean Triples................................................. 13 3 Pythagorean Triples and the Unit Circle............................... 21 4 Sums of Higher Powersand Fermat’s Last Theorem.................... 26 5 Divisibility and the Greatest Common Divisor......................... 30 6 Linear Equations and the Greatest Common Divisor.................... 37 7 Factorization and the Fundamental Theorem of Arithmetic.............. 46 8 Congruences........................................................ 55 9 Congruences,Powers, and Fermat’s Little Theorem..................... 65 10 Congruences,Powers, and Euler’s Formula............................ 71 11 Euler’s Phi Function and the Chinese Remainder Theorem.............. 75 12 Prime Numbers..................................................... 83 13 Counting Primes.................................................... 90 14 Mersenne Primes.................................................... 96 15 Mersenne Primes and Perfect Numbers............................... 101 16 Powers Modulom and Successive Squaring........................... 111 17 Computing k th Roots Modulom ..................................... 118 18 Powers,Roots,and“Unbreakable”Codes............................ 123 19 Primality Testing and Carmichael Numbers........................... 129 20 Squares Modulo p .................................................. 141 21 Is.1 a Square Modulo p?Is 2?..................................... 148 22 Quadratic Reciprocity.............................................. 159 23 Proof of Quadratic Reciprocity...................................... 171 24 Which Primes Are Sums of Two Squares?............................ 181 25 Which Numbers Are Sums of Two Squares?.......................... 193 26 As Easyas One,Two,Three........................................ 199 27 Euler’s Phi Function and Sums of Divisors........................... 206 28 Powers Modulo p and Primitive Roots............................... 211 29 Primitive Roots and Indices......................................... 224 30 The Equation X 4+Y 4=Z 4 .......................................... 231 31 Square–Triangular Numbers Revisited............................... 236 32 Pell’sEquation .................................................... 245 33 Diophantine Approximation......................................... 251 34 Diophantine Approximation and Pell’s Equation...................... 260 35 Numb
鄂公網(wǎng)安備 42010302001612號 