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Introduction to Number Theory

American Mathematical Society
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Thursday, July 27, 2006
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Trygve Nagell
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  • Divisibility: 1.1 Divisors; 1.2 Remainders; 1.3 Primes; 1.4 The fundamental theorem; 1.5 Least common multiple and greatest common divisor; 1.6 Moduls, rings and fields; 1.7 Euclid's algorithm; 1.8 Relatively prime numbers. Euler's $\varphi$-function; 1.9 Arithmetical functions; 1.10 Diophantine equations of the first degree; 1.11 Lattice points and point lattices; 1.12 Irrational numbers; 1.13 Irrationality of the numbers $e$ and $\pi$; Exercises (1--40)
  • On the Distribution of Primes: 2.14 Some lemmata; 2.15 General remarks. The sieve of Eratosthenes; 2.16 The function $\pi(x)$; 2.17 Some elementary results on the distribution of primes; 2.18 Other problems and results concerning primes
  • Theory of Congruences: 3.19 Definitions and fundamental properties; 3.20 Residue classes and residue systems; 3.21 Fermat's theorem and its generalization by Euler; 3.22 Algebraic congruences and functional congruences; 3.23 Linear congruences; 3.24 Algebraic congruences to a prime modulus; 3.25 Prime divisors of integral polynomials; 3.26 Algebraic congruences to a composite modulus; 3.27 Algebraic congruences to a prime- power modulus; 3.28 Numerical examples of solution of algebraic congruences; 3.29 Divisibility of integral polynomials with regard to a prime modulus; 3.30 Wilson's theorem and its generalization; 3.31 Exponent of an integer modulo $n$; 3.32 Moduli having primitive roots; 3.33 The index calculus; 3.34 Power residues. Binomial congruences; 3.35 Polynomials representing integers; 3.36 Thue's remainder theorem and its generalization by Scholz; Exercises (41-89)
  • Theory of Quadratic Residues: 4.37 The general quadratic congruence; 4.38 Euler's criterion and Legendre's symbol; 4.39 On the solvability of the congruences $x^2\equiv\pm 2\, (\text{mod }p)$; 4.40 Gauss's lemma; 4.41 The quadratic reciprocity law; 4.42 Jacobi's symbol and the generalization of the reciprocity law; 4.43 The prime divisors of quadratic polynomials; 4.44 Primes in special arithmetical progressions
  • Arithmetical Properties of the Roots of Unity: 5.45 The roots of unity; 5.46 The cyclotomic polynomial; 5.47 Irreducibility of the cyclotomic polynomial; 5.48 The prime divisors of the cyclotomic polynomial; 5.49 A theorem of Bauer on the prime divisors of certain polynomials; 5.50 On the primes of the form $n y-1$; 5.51 Some trigonometrical products; 5.52 A polynomial identity of Gauss; 5.53 The Gaussian sums; Exercises (90-122)
  • Diophantine Equations of the Second Degree: 6.54 The representation of integers as sums of integral squares; 6.55 Bachet's theorem; 6.56 The Diophantine equation $x^2-Dy^2=1$; 6.57 The Diophantine Equation $x^2-Dy^2=-1$; 6.58 The Diophantine equation $u^2-Dv^2=C$; 6.59 Lattice points on conics; 6.60 Rational points in the plane and on conics; 6.61 The Diophantine equation $ax^2+by^2+cz^2=0$
  • Diophantine Equations of Higher Degree: 7.62 Some Diophantine equations of the fourth degree with three unknowns; 7.63 The Diophantine equation $2x^4-y^4=z^2$; 7.64 The quadratic fields $K(\sqrt{-1}), K(\sqrt{-2})$ and $K(\sqrt{-3})$; 7.65 The Diophantine equation $\xi^3+\eta^3+\zeta^3=0$ and analogous equations; 7.66 Diophantine equations of the third degree with an infinity of solutions; 7.67 The Diophantine equation $x^7+y^7+z^7=0$; 7.68 Fermat's last theorem; 7.69 Rational points on plane algebraic curves. Mordell's theorem; 7.70 Lattice points on plane algebraic curves. Theorems of Thue and Siegel; Exercises (123-171)
  • The Prime Number Theorem: 8.71 Lemmata on the order of magnitude of some finite sums; 8.72 Lemmata on the Möbius function and some related functions; 8.73 Further lemmata. Proof of Selberg's formula; 8.74 An elementary proof of the prime number theorem; Exercises (172-180)
  • Table of primitive roots
  • Fundamental solutions of equations $x^2-Dy^2=\pm 1$
  • Name index
  • Subject index
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Wednesday, July 15, 2009