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Number Theory in the Spirit of Liouville

$61.99 (P)

Part of London Mathematical Society Student Texts

  • Date Published: November 2010
  • availability: Available
  • format: Paperback
  • isbn: 9780521175623

$ 61.99 (P)

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About the Authors
  • Joseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville's ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.

    • Demonstrates that some analytic formulae in number theory can be proved in an elementary arithmetic manner
    • Motivates students to do their own research
    • Includes an extensive bibliography
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    Reviews & endorsements

    "... a fascinating exploration and reexamination of both Liouville's identities and "elementary" methods, providing revealing connections to modern techniques and proofs. Overall, the work contributes significantly to both number theory and the history of mathematics."
    J. Johnson, Choice Magazine

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    Product details

    • Date Published: November 2010
    • format: Paperback
    • isbn: 9780521175623
    • length: 306 pages
    • dimensions: 227 x 151 x 16 mm
    • weight: 0.45kg
    • contains: 275 exercises
    • availability: Available
  • Table of Contents

    1. Joseph Liouville (1809–1888)
    2. Liouville's ideas in number theory
    3. The arithmetic functions σk(n), σk*(n), dk,m(n) and Fk(n)
    4. The equation i2 + jk = n
    5. An identity of Liouville
    6. A recurrence relation for σ*(n)
    7. The Girard–Fermat theorem
    8. A second identity of Liouville
    9. Sums of two, four and six squares
    10. A third identity of Liouville
    11. Jacobi's four squares formula
    12. Besge's formula
    13. An identity of Huard, Ou, Spearman and Williams
    14. Four elementary arithmetic formulae
    15. Some twisted convolution sums
    16. Sums of two, four, six and eight triangular numbers
    17. Sums of integers of the form x2+xy+y2
    18. Representations by x2+y2+z2+2t2, x2+y2+2z2+2t2 and x2+2y2+2z2+2t2
    19. Sums of eight and twelve squares
    20. Concluding remarks

  • Author

    Kenneth S. Williams, Carleton University, Ottawa
    Kenneth S. Williams is currently Professor Emeritus and Distinguished Research Professor of Mathematics at Carleton University, Ottawa.

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