Book contents
- Frontmatter
- Contents
- Preface
- Notation
- 1 Joseph Liouville (1809–1882)
- 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
- References
- Index
Preface
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- Preface
- Notation
- 1 Joseph Liouville (1809–1882)
- 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
- References
- Index
Summary
In a series of eighteen papers published between the years 1858 and 1865 the French mathematician Joseph Liouville (1809–1882) introduced a powerful new method into elementary number theory. Liouville's idea was to give a number of elementary (but not simple to prove) identities from which flowed many number-theoretic results by specializing the functions involved in the formulae.
Although Liouville's ideas are now 150 years old, they still do not usually form part of a standard course in elementary number theory. Moreover there is no book in English devoted entirely to Liouville's method, and, although some elementary number theory texts devote a chapter to Liouville's ideas, most do not. In this book we hope to remedy this situation by providing a gentle introduction to Liouville's method. We will not give a comprehensive treatment of all of Liouville's identities but rather give a sufficient number of his identities in order to provide elementary arithmetic proofs of such number-theoretic results as the Girard-Fermat theorem, a recurrence relation for the sum of divisors function, Lagrange's theorem, Legendre's formula for the number of representations of a nonnegative integer as the sum of four triangular numbers, Jacobi's formula for the number of representations of a positive integer as the sum of eight squares, and many others. We will also treat some of the more recent results that have been obtained using Liouville's ideas.
- Type
- Chapter
- Information
- Number Theory in the Spirit of Liouville , pp. xi - xiiPublisher: Cambridge University PressPrint publication year: 2010