Looking for an examination copy?
If you are interested in the title for your course we can consider offering an examination copy. To register your interest please contact firstname.lastname@example.org providing details of the course you are teaching.
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.Read more
- 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
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
Not yet reviewed
Be the first to review
Review was not posted due to profanity×
- 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
Sorry, this resource is locked
Please register or sign in to request access. If you are having problems accessing these resources please email email@example.comRegister Sign in
You are now leaving the Cambridge University Press website. Your eBook purchase and download will be completed by our partner www.ebooks.com. Please see the permission section of the www.ebooks.com catalogue page for details of the print & copy limits on our eBooks.Continue ×
Are you sure you want to delete your account?
This cannot be undone.
Thank you for your feedback which will help us improve our service.
If you requested a response, we will make sure to get back to you shortly.×