About our author Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and 7 books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography. He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.

• Preface Flowchart of Chapter Dependencies Introduction What Is Number Theory?Pythagorean TriplesPythagorean Triples and the Unit CircleSums of Higher Powers and Fermat's Last TheoremDivisibility and the Greatest Common DivisorLinear Equations and the Greatest Common DivisorFactorization and the Fundamental Theorem of ArithmeticCongruencesCongruences, Powers, and Fermat's Little TheoremCongruences, Powers, and Euler's FormulaEuler's Phi Function and the Chinese Remainder TheoremPrime NumbersCounting PrimesMersenne PrimesMersenne Primes and Perfect NumbersPowers Modulo m and Successive SquaringComputing kth Roots Modulo mPowers, Roots, and “Unbreakable” CodesPrimality Testing and Carmichael NumbersSquares Modulo pIs -1 a Square Modulo p? Is 2?Quadratic ReciprocityProof of Quadratic ReciprocityWhich Primes Are Sums of Two Squares?Which Numbers Are Sums of Two Squares?As Easy as One, Two, ThreeEuler's Phi Function and Sums of DivisorsPowers Modulo p and Primitive RootsPrimitive Roots and IndicesThe Equation X4 + Y4 = Z4Square - Triangular Numbers RevisitedPell's EquationDiophantine ApproximationDiophantine Approximation and Pell's EquationNumber Theory and Imaginary NumbersThe Gaussian Integers and Unique FactorizationIrrational Numbers and Transcendental NumbersBinomial Coefficients and Pascal's TriangleFibonacci's Rabbits and Linear Recurrence SequencesOh, What a Beautiful FunctionCubic Curves and Elliptic CurvesElliptic Curves with Few Rational PointsPoints on Elliptic Curves Modulo pTorsion Collections Modulo p and Bad PrimesDefect Bounds and Modularity PatternsElliptic Curves and Fermat's Last TheoremThe Topsy-Turvey World of Continued Fractions [online]Continued Fractions, Square Roots, and Pell's Equation [online]Generating Functions [online]Sums of Powers [online]Further Reading Index A. Factorization of Small Composite Integers [online] B. A List of Primes [online]

About our author Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and 7 books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography. He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.

• Preface Flowchart of Chapter Dependencies Introduction What Is Number Theory?Pythagorean TriplesPythagorean Triples and the Unit CircleSums of Higher Powers and Fermat's Last TheoremDivisibility and the Greatest Common DivisorLinear Equations and the Greatest Common DivisorFactorization and the Fundamental Theorem of ArithmeticCongruencesCongruences, Powers, and Fermat's Little TheoremCongruences, Powers, and Euler's FormulaEuler's Phi Function and the Chinese Remainder TheoremPrime NumbersCounting PrimesMersenne PrimesMersenne Primes and Perfect NumbersPowers Modulo m and Successive SquaringComputing kth Roots Modulo mPowers, Roots, and “Unbreakable” CodesPrimality Testing and Carmichael NumbersSquares Modulo pIs -1 a Square Modulo p? Is 2?Quadratic ReciprocityProof of Quadratic ReciprocityWhich Primes Are Sums of Two Squares?Which Numbers Are Sums of Two Squares?As Easy as One, Two, ThreeEuler's Phi Function and Sums of DivisorsPowers Modulo p and Primitive RootsPrimitive Roots and IndicesThe Equation X4 + Y4 = Z4Square - Triangular Numbers RevisitedPell's EquationDiophantine ApproximationDiophantine Approximation and Pell's EquationNumber Theory and Imaginary NumbersThe Gaussian Integers and Unique FactorizationIrrational Numbers and Transcendental NumbersBinomial Coefficients and Pascal's TriangleFibonacci's Rabbits and Linear Recurrence SequencesOh, What a Beautiful FunctionCubic Curves and Elliptic CurvesElliptic Curves with Few Rational PointsPoints on Elliptic Curves Modulo pTorsion Collections Modulo p and Bad PrimesDefect Bounds and Modularity PatternsElliptic Curves and Fermat's Last TheoremThe Topsy-Turvey World of Continued Fractions [online]Continued Fractions, Square Roots, and Pell's Equation [online]Generating Functions [online]Sums of Powers [online]Further Reading Index A. Factorization of Small Composite Integers [online] B. A List of Primes [online]

About our author Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and 7 books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography. He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.

• Preface Flowchart of Chapter Dependencies Introduction What Is Number Theory?Pythagorean TriplesPythagorean Triples and the Unit CircleSums of Higher Powers and Fermat's Last TheoremDivisibility and the Greatest Common DivisorLinear Equations and the Greatest Common DivisorFactorization and the Fundamental Theorem of ArithmeticCongruencesCongruences, Powers, and Fermat's Little TheoremCongruences, Powers, and Euler's FormulaEuler's Phi Function and the Chinese Remainder TheoremPrime NumbersCounting PrimesMersenne PrimesMersenne Primes and Perfect NumbersPowers Modulo m and Successive SquaringComputing kth Roots Modulo mPowers, Roots, and “Unbreakable” CodesPrimality Testing and Carmichael NumbersSquares Modulo pIs -1 a Square Modulo p? Is 2?Quadratic ReciprocityProof of Quadratic ReciprocityWhich Primes Are Sums of Two Squares?Which Numbers Are Sums of Two Squares?As Easy as One, Two, ThreeEuler's Phi Function and Sums of DivisorsPowers Modulo p and Primitive RootsPrimitive Roots and IndicesThe Equation X4 + Y4 = Z4Square - Triangular Numbers RevisitedPell's EquationDiophantine ApproximationDiophantine Approximation and Pell's EquationNumber Theory and Imaginary NumbersThe Gaussian Integers and Unique FactorizationIrrational Numbers and Transcendental NumbersBinomial Coefficients and Pascal's TriangleFibonacci's Rabbits and Linear Recurrence SequencesOh, What a Beautiful FunctionCubic Curves and Elliptic CurvesElliptic Curves with Few Rational PointsPoints on Elliptic Curves Modulo pTorsion Collections Modulo p and Bad PrimesDefect Bounds and Modularity PatternsElliptic Curves and Fermat's Last TheoremThe Topsy-Turvey World of Continued Fractions [online]Continued Fractions, Square Roots, and Pell's Equation [online]Generating Functions [online]Sums of Powers [online]Further Reading Index A. Factorization of Small Composite Integers [online] B. A List of Primes [online]