staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Erin McAfee; Kenneth S. Williams
An arithmetic formula of Liouville
Journal de théorie des nombres de Bordeaux, 18 no. 1 (2006), p. 223-239, doi: 10.5802/jtnb.541
Article PDF | Reviews MR 2245883 | Zbl 05070455

Résumé - Abstract

An elementary proof is given of an arithmetic formula, which was stated but not proved by Liouville. An application of this formula yields a formula for the number of representations of a positive integer as the sum of twelve triangular numbers.

Bibliography

[1] L. E. Dickson, History of the Theory of Numbers. Vol. 1 (1919), Vol. 2 (1920), Vol. 3 (1923), Carnegie Institute of Washington, reprinted Chelsea, NY, 1952.  Zbl 0958.11500
[2] J. G. Huard, Z. M. Ou, B. K. Spearman, K. S. Williams, Elementary evaluation of certain convolution sums involving divisor functions. Number Theory for the Millenium II, 229–274. M. A. Bennett et al., editors, A. K. Peters Ltd, Natick, Massachusetts, 2002.  MR 1956253 |  Zbl 1062.11005
[3] J. Liouville, Sur quelques formule générales qui peuvent être utiles dans la théorie des nombres. (premier article) 3 (1858), 143–152; (deuxième article) 3 (1858), 193–200; (troisième article) 3 (1858), 201-208; (quatrième article) 3 (1858), 241–250; (cinquième article) 3 (1858), 273–288; (sixième article) 3 (1858), 325–336; (septième article) 4 (1859), 1–8; (huitième article) 4 (1859), 73–80; (neuvième article) 4 (1859), 111–120; (dixième article) 4 (1859), 195–204. (onzième article) 4 (1859), 281–304; (douzième article) 5 (1860), 1–8; (treizième article) 9 (1864), 249–256; (quatorzième article) 9 (1864), 281–288; (quinzième article) 9 (1864), 321–336; (seizième article) 9 (1864), 389–400; (dix-septième article) 10 (1865), 135–144; (dix-huitième article) 10 (1865), 169–176.
[4] E. McAfee, A three term arithmetic formula of Liouville type with application to sums of six squares. M. Sc. thesis, Carleton University, Ottawa, Canada, 2004.
[5] P. S. Nasimoff, Applications to the Theory of Elliptic Functions to the Theory of Numbers. Moscow, 1884.  JFM 54.0196.01
[6] K. Ono, S. Robins, P. T. Wahl, On the representation of integers as sums of triangular numbers. Aequationes Math. 50 (1995), 73–94.  MR 1336863 |  Zbl 0828.11057