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Mathematics of Computation

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Elliptic binomial diophantine equations

Authors: Roelof J. Stroeker and Benjamin M. M. de Weger
Journal: Math. Comp. 68 (1999), 1257-1281
MSC (1991): Primary 11D25, 11G05; Secondary 11B65, 14H52
Published electronically: February 23, 1999
MathSciNet review: 1622097
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Abstract: The complete sets of solutions of the equation $ \binom{n}{k} = \binom{m}{\ell} $ are determined for the cases $ (k,\ell) = (2,3) $, $ (2,4) $, $ (2,6) $, $ (2,8) $, $ (3,4) $, $ (3,6) $, $ (4,6) $, $ (4,8) $. In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ($ (k,\ell) = (3,6) $) we had to devise a new variant of the method.

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Additional Information

Roelof J. Stroeker
Affiliation: Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands

Benjamin M. M. de Weger
Affiliation: Sportsingel 30, 2924 XN Krimpen aan den ijssel, The Neterlands

Keywords: Diophantine equation, elliptic curve, binomial coefficient
Received by editor(s): October 16, 1997
Published electronically: February 23, 1999
Additional Notes: The second author’s research was supported by the Netherlands Mathematical Research Foundation SWON with financial aid from the Netherlands Organization for Scientific Research NWO
Article copyright: © Copyright 1999 American Mathematical Society