Elliptic binomial diophantine equations

Stroeker, Roelof; de Weger, Benjamin

doi:10.1090/S0025-5718-99-01047-9

Elliptic binomial diophantine equations
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by Roelof J. Stroeker and Benjamin M. M. de Weger PDF

Math. Comp. 68 (1999), 1257-1281 Request permission

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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