An arithmetic formula of Liouville

Erin McAfee[1]; Kenneth S. Williams[1]

  • [1] School of Mathematics and Statistics Carleton University Ottawa, Ontario, Canada K1S 5B6

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 223-239
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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McAfee, Erin, and Williams, Kenneth S.. "An arithmetic formula of Liouville." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 223-239. <http://eudml.org/doc/249656>.

@article{McAfee2006,
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.},
affiliation = {School of Mathematics and Statistics Carleton University Ottawa, Ontario, Canada K1S 5B6; School of Mathematics and Statistics Carleton University Ottawa, Ontario, Canada K1S 5B6},
author = {McAfee, Erin, Williams, Kenneth S.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {arithmetic formula of Liouville; triangular number; divisor function; linear Diophantine equation},
language = {eng},
number = {1},
pages = {223-239},
publisher = {Université Bordeaux 1},
title = {An arithmetic formula of Liouville},
url = {http://eudml.org/doc/249656},
volume = {18},
year = {2006},
}

TY - JOUR
AU - McAfee, Erin
AU - Williams, Kenneth S.
TI - An arithmetic formula of Liouville
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 223
EP - 239
AB - 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.
LA - eng
KW - arithmetic formula of Liouville; triangular number; divisor function; linear Diophantine equation
UR - http://eudml.org/doc/249656
ER -

References

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  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. Zbl0958.11500
  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. Zbl1062.11005MR1956253
  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. 
  6. K. Ono, S. Robins, P. T. Wahl, On the representation of integers as sums of triangular numbers. Aequationes Math. 50 (1995), 73–94. Zbl0828.11057MR1336863

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