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Higher arithmetic : an algorithmic introduction to number theory / Harold M. Edwards.

By: Edwards, Harold M.
Material type: materialTypeLabelBookSeries: Student mathematical library, v. 45.Publisher: Providence, R.I. : American Mathematical Society, c2008Description: xii, 210 p. : ill. ; 22 cm. + pbk.ISBN: 9780821844397 .Subject(s): Number theory | ArithmeticDDC classification: 512.7
Contents:
Numbers -- The problem A() + B = () -- Congruences -- Double congruences and euclidean algorithm -- The augmented euclidean algorithm -- Simultaneous congruences -- The fundamental theorem of arithmetic -- Exponentiation and orders -- Euler's function -- Finding the order of a mod c -- Primality testing -- The RSA cipher system -- Primitive roots mod p -- polynomials -- Tables of indices mod p -- Brahmagupta's formula and hypernumbers -- Modules of hypernumbers -- A canonical form for modules of hypernumbers -- Solution of A() + B = () -- Proof of the theorem of chapter 19 -- Euler's remarkable discovery -- Stable modules -- Equivalence of modules -- Signatures of equivalence classes -- The main theorem -- Modules that become principal when squared -- The possible signatures for certain values of A -- The law of quadratic reciprocity -- Proof of the main theorem -- The theory of binary quadratic forms -- Composition of binary quadratic forms.

Enhanced descriptions from Syndetics:

Although number theorists have sometimes shunned and even disparaged computation in the past, today's applications of number theory to cryptography and computer security demand vast arithmetical computations. These demands have shifted the focus of studies in number theory and have changed attitudes toward computation itself.

Bibliography: (page 207) and index.

Numbers -- The problem A() + B = () -- Congruences -- Double congruences and euclidean algorithm -- The augmented euclidean algorithm -- Simultaneous congruences -- The fundamental theorem of arithmetic -- Exponentiation and orders -- Euler's function -- Finding the order of a mod c -- Primality testing -- The RSA cipher system -- Primitive roots mod p -- polynomials -- Tables of indices mod p -- Brahmagupta's formula and hypernumbers -- Modules of hypernumbers -- A canonical form for modules of hypernumbers -- Solution of A() + B = () -- Proof of the theorem of chapter 19 -- Euler's remarkable discovery -- Stable modules -- Equivalence of modules -- Signatures of equivalence classes -- The main theorem -- Modules that become principal when squared -- The possible signatures for certain values of A -- The law of quadratic reciprocity -- Proof of the main theorem -- The theory of binary quadratic forms -- Composition of binary quadratic forms.

Table of contents provided by Syndetics

  • Introduction to Number Theory
  • Numbers The problem $A\square + B = \square$ Congruences
  • Double congruences and the Euclidean algorithm
  • The augmented Euclidean algorithm
  • Simultaneous congruences
  • The fundamental theorem of arithmetic
  • Exponentiation and orders
  • Euler's $\phi$-function
  • Finding the order of $a\bmod c$ Primality testing
  • The RSA cipher system
  • Primitive roots $\bmod\ p$
  • Polynomials Tables of indices $\bmod\ p$
  • Brahmagupta's formula and hypernumbers
  • Modules of hypernumbers
  • A canonical form for modules of hypernumbers
  • Solution of $A\square + B = \square$
  • Proof of the theorem of Chapter 19
  • Euler's remarkable discovery
  • Stable modules
  • Equivalence of modules
  • Signatures of equivalence classes
  • The main theorem
  • Which modules become principal when squared?
  • The possible signatures for certain values of $A$
  • The law of quadratic reciprocity
  • Proof of the Main Theorem
  • The theory of binary quadratic forms
  • Composition of binary quadratic forms
  • Cycles of stable modules
  • Answers to exercises
  • Bibliography
  • Index

Reviews provided by Syndetics

CHOICE Review

Edwards (emer., NYU), already famous for his historically informed expositions in his previous works (e.g., Fermat's Last Theorem, 1977; Riemann's Zeta Function, CH, Oct'74), sets out the details concerning the solution of a major problem of Diophantine analysis: given numbers A and B, find all pairs of squares such that A times the first plus B equals the second. Clean and elegant in the way it communicates with the reader, the mathematical spirit of this book remains very close to that of C. F. Gauss in his 1801 Disquisitiones Arithmeticae, almost as though Gauss had revised that classic for 21st-century readers (with high-speed computers). So, as a tour de force, Edwards eschews (mostly) the tools unavailable to Gauss, including modern algebraic number theory and even the basic apparatus of abstract algebra. He even avoids the use of negative numbers. One gets the pleasure of Gauss' insights without battling Gauss' obscurities. But the student who masters this book may have some trouble making the transition to modern language, so one only wishes Edwards had supplied an appendix designed to help students efficiently apply the insights gleaned herein to future study. Summing Up: Recommended. Lower- and upper-division undergraduates, researchers, faculty, and general readers. D. V. Feldman University of New Hampshire

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