Includes bibliographical references.

Machine generated contents note: I.The Division Algorithm -- 1.The Origin of Natural Numbers -- Perfect Numbers and Amicable Numbers -- 2.The Mystery of Natural Numbers -- Mosaic Geometry and Euler's Characteristic -- 3.The Division Algorithm -- Mersenne Primes and Fermat Primes -- 4.The Greatest Common Divisor -- Graham's Conjecture -- 5.The Fundamental Theorem of Arithmetic -- Hilbert's 8th Problem -- Exercises -- II.The Concept of Congruence -- 6.The Concept of Congruence -- Gauss' Disquisitiones Arithmeticae -- 7.Residue Classes and Residue Systems -- Function [x] and the 3x + 1 Problem -- 8.The Fermat -- Euler Theorem -- The Euler Number and the Euler Prime -- 9.Fractions Expressed as Repeating Decimals -- Mobius' Function -- 10.An Application to Cryptology -- The Generalized Euler Function -- Exercises 2 -- III.Congruences -- 11.Qin Jiushao's Theorem -- Fibonacci's Rabbits -- 12.Wilson's Theorem -- A Theorem that Gauss Did Not Prove -- 13.The Diophantine Equation Note continued: The Pythagorean Triple -- 14.Lucas' Congruence -- Covering Systems -- 15.The Truth of Primes -- The Chain of Primes or Composite Numbers -- Exercises 3 -- IV.Quadratic Residues -- 16.Quadratic Congruences -- Integers in the Gaussian Ring -- 17.The Legendre Symbol -- Representing Integers as the Sum of Squares -- 18.The Law of Quadratic Reciprocity -- N-Gonal Numbers and Fermat -- 19.The Jacobi Symbol -- The Hadamard Matrix and Hadamard's Conjecture -- 20.Congruences Modulo a Composite -- Constructibility of the Regular 17-Gon -- Exercises 4 -- V.Nth Power Residues -- 21.Definition of Orders -- Egyptian Fractions -- 22.The Existence of Primitive Roots -- Artin's Conjecture -- 23.The Nth Power Residue -- Pell's Equation -- 24.The Case of Composite Modulus -- Diophantine Arrays -- 25.The Dirichlet Character -- Three Special Exponent Sums -- Exercises 5 -- VI.Congruences Modulo Integer Powers -- 26.Bernoulli Numbers and Bernoulli Polynomials Note continued: The Kummer Congruence -- 27.The Wolstenholme Theorem -- Elliptic Curves -- 28.Lehmer's Congruence -- The abc Conjecture -- 29.Morley's Theorem and Jacobstahl's Theorem -- Automorphic Forms and Modular Forms -- 30.Congruences on Harmonic Sums -- Multinomial Coefficients of Non-Powers -- VII.Additive and Multiplicative Number Theory -- 31.New Waring's Problem -- 32.New Fermat's Last Theorem -- 33.Euler's Conjecture -- 34.The F-Perfect Number Problem -- 35.A New Congruent Number Problem -- 36.The ABCD Equation.

Natural numbers are the oldest human inventions. This volume describes their nature, laws, history and current status. The first five chapters contain not only the basics of elementary number theory for the convenience of teaching and continuity of reading, but also many latest research results.

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