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
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topMcAfee, 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
top- 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
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- 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.
- 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.
- P. S. Nasimoff, Applications to the Theory of Elliptic Functions to the Theory of Numbers. Moscow, 1884.
- 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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