Elliptic binomial diophantine equations

The complete sets of solutions of the equation ( n k ) = ( m l ) are determined for the cases (k, l) = (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, l) = (3, 6)) we had to devise a new variant of the method.

[1]  De Weger.  A binomial diophantine equation , 1996 .

[2]  J. Coates,et al.  Integer points on curves of genus 1 , 1970, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  J. Cremona.  Algorithms for Modular Elliptic Curves , 1992 .

[4]  Э. Т. Аванесов.  Решение одной проблемы фигурных чисел , 1967 .

[5]  T. Nagell.  Sur les propriétés arithmétiques des cubiques planes du premier genre , 1929 .

[6]  De Weger,et al.  de Weger: On the practical solution of the Thue equation , 1989 .

[7]  L. Mordell,et al.  Diophantine equations , 1969 .

[8]  Attila Pethő,et al.  Computing integral points on elliptic curves , 1994 .

[9]  B. Weger.  Equal Binomial Coefficients: Some Elementary Considerations , 1997 .

[10]  Joseph H. Silverman,et al.  The arithmetic of elliptic curves , 1986, Graduate texts in mathematics.

[11]  G. Hanrot,et al.  Solving superelliptic Diophantine equations by Baker's method , 1998, Compositio Mathematica.

[12]  Roel J. Stroeker,et al.  On the Elliptic Logarithm Method for Elliptic Diophantine Equations: Reflections and an Improvement , 1999, Exp. Math..

[13]  S. David.  Minorations de formes linéaires de logarithmes elliptiques , 1995 .

[14]  G. Faltings.  Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , 1983 .

[15]  Nikos Tzanakis,et al.  Solving Elliptic Diophantine Equations by estimating Linear Forms in Elliptic Logarithms. The case o , 1994 .

[16]  B. D. Weger,et al.  Solving elliptic diophantine equations: the general cubic case , 1999 .

[17]  S. Siksek.  Infinite Descent on Elliptic Curves , 1995 .

[18]  J. Kellett.  London , 1914, The Hospital.

[19]  L. Mordell.  On the integer solutions of $y(y+1)=x(x+1)(x+2)$. , 1963 .

[20]  Joseph H. Silverman,et al.  The difference between the Weil height and the canonical height on elliptic curves , 1990 .

[21]  Wolfgang M. Schmidt,et al.  Integer points on curves of genus 1 , 1992 .

[22]  N. Tzanakis,et al.  Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms , 1994 .

[23]  Nigel P. Smart,et al.  S-integral points on elliptic curves , 1994, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  N. Smart.  Integral points on elliptic curves , 1998 .