Introduction to Number Theory, 2nd Edit... | WHSmith Books
Introduction to Number Theory, 2nd Edition (Textbooks in Mathematics 2nd New edition)

Introduction to Number Theory, 2nd Edition (Textbooks in Mathematics 2nd New edition)

By: Anthony Vazzana (author), David Garth (author)Hardback

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Description

Introduction to Number Theory is a classroom-tested, student-friendly text that covers a diverse array of number theory topics, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments such as cryptography, the theory of elliptic curves, and the negative solution of Hilbert's tenth problem. The authors illustrate the connections between number theory and other areas of mathematics, including algebra, analysis, and combinatorics. They also describe applications of number theory to real-world problems, such as congruences in the ISBN system, modular arithmetic and Euler's theorem in RSA encryption, and quadratic residues in the construction of tournaments. Ideal for a one- or two-semester undergraduate-level course, this Second Edition: Features a more flexible structure that offers a greater range of options for course designAdds new sections on the representations of integers and the Chinese remainder theoremExpands exercise sets to encompass a wider variety of problems, many of which relate number theory to fields outside of mathematics (e.g., music)Provides calculations for computational experimentation using SageMath, a free open-source mathematics software system, as well as Mathematica (R) and Maple (TM), online via a robust, author-maintained websiteIncludes a solutions manual with qualifying course adoption By tackling both fundamental and advanced subjects-and using worked examples, numerous exercises, and popular software packages to ensure a practical understanding-Introduction to Number Theory, Second Edition instills a solid foundation of number theory knowledge.

About Author

Martin Erickson (1963-2013) received his Ph.D in mathematics in 1987 from the University of Michigan, Ann Arbor, USA, studying with Thomas Frederick Storer. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, when he was twenty-four years old, and remained there for the rest of his life. Professor Erickson authored and coauthored several mathematics books, including the first edition of Introduction to Number Theory (CRC Press, 2007), Pearls of Discrete Mathematics (CRC Press, 2010), and A Student's Guide to the Study, Practice, and Tools of Modern Mathematics (CRC Press, 2010). Anthony Vazzana received his Ph.D in mathematics in 1998 from the University of Michigan, Ann Arbor, USA. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, in 1998. In 2000, he was awarded the Governor's Award for Excellence in Teaching and was selected as the Educator of the Year. In 2002, he was named the Missouri Professor of the Year by the Carnegie Foundation for the Advancement of Teaching and the Council for Advancement and Support of Education. David Garth received his Ph.D in mathematics in 2000 from Kansas State University, Manhattan, USA. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, in 2000. In 2005, he was awarded the Golden Apple Award from Truman State University's Theta Kappa chapter of the Order of Omega. His areas of research include analytic and algebraic number theory, especially Pisot numbers and their generalizations, and Diophantine approximation.

Contents

IntroductionWhat is number theory?The natural numbersMathematical inductionNotesThe Peano axioms DivisibilityBasic definitions and propertiesThe division algorithmRepresentations of integers Greatest Common DivisorGreatest common divisorThe Euclidean algorithmLinear Diophantine equationsNotesEuclidThe number of steps in the Euclidean algorithmGeometric interpretation of the equation ax + by = c PrimesThe sieve of EratosthenesThe fundamental theorem of arithmeticDistribution of prime numbersNotesEratosthenesNonunique factorization and Fermat's last theorem CongruencesResidue classesLinear congruencesApplication: Check digits and the ISBN-10 systemThe Chinese remainder theorem Special CongruencesFermat's theoremEuler's theoremWilson's theoremNotesLeonhard Euler Primitive RootsOrder of an element mod nExistence of primitive rootsPrimitive roots modulo compositesApplication: Construction of the regular 17-gonNotesGroupsStraightedge and compass constructions CryptographyMonoalphabetic substitution ciphersThe Pohlig-Hellman cipherThe Massey-Omura exchangeThe RSA algorithmNotesComputing powers mod pRSA cryptography Quadratic ResiduesQuadratic congruencesQuadratic residues and nonresiduesQuadratic reciprocityThe Jacobi symbolNotesCarl Friedrich Gauss Applications of Quadratic ResiduesApplication: Construction of tournamentsConsecutive quadratic residues and nonresiduesApplication: Hadamard matrices Sums of SquaresPythagorean triplesGaussian integersFactorization of Gaussian integersLagrange's four squares theoremNotesDiophantus Further Topics in Diophantine EquationsThe case n = 4 in Fermat's last theoremPen's equationThe abc conjectureNotesPierre de FermatThe p-adic numbers Continued FractionsFinite continued fractionsInfinite continued fractionsRational approximation of real numbersNotesContinued fraction expansion of eContinued fraction expansion of tan xSrinivasa Ramanujan Continued Fraction Expansions of Quadratic IrrationalsPeriodic continued fractionsContinued fraction factorizationContinued fraction solution of Pen's equationNotesThree squares and triangular numbersHistory of Pen's equation Arithmetic FunctionsPerfect numbersThe group of arithmetic functionsMobius inversionApplication: Cyclotomic polynomialsPartitions of an integerNotesThe lore of perfect numbersPioneers of integer partitions Large PrimesFermat numbersMersenne numbersPrime certificatesFinding large primes Analytic Number TheorySum of reciprocals of primesOrders of growth of functionsChebyshev's theoremBertrand's postulateThe prime number theoremThe zeta function and the Riemann hypothesisDirichlet's theoremNotesPaul Erdos Elliptic CurvesCubic curvesIntersections of lines and curvesThe group law and addition formulasSums of two cubesElliptic curves mod pEncryption via elliptic curvesElliptic curve method of factorizationFermat's last theoremNotesProjective spaceAssociativity of the group law

Product Details

  • ISBN13: 9781498717496
  • Format: Hardback
  • Number Of Pages: 414
  • ID: 9781498717496
  • weight: 748
  • ISBN10: 1498717497
  • edition: 2nd New edition

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