Effective solution of two simultaneous Pell equations by the elliptic logarithm method

A nonextendable Diophantine quadruple arising from a triple of Lucas numbers

“…S. P. Mohanty and A. M. S. Ramasamy [1985] introduced the concept of the characteristic number of two simultaneous Pell's equations and solved the system U 2 − 5V 2 = −4 and Z 2 − 12V 2 = −11. N. Tzanakis [2002] gave a method in for solving a system of Pell's equations using elliptic logarithms, and earlier [1993] described various methods available in the literature for finding out the common solutions of a system of Pell's equations. (For a history of numbers with the Diophantine property, one may refer to [Ramasamy 2007].…”

On the number of solutions to systems of Pell equations

“…The output of a code written in PARI/GP [5] is that the inequality (27) does not hold for j 3 > 167. Our gap principles and the obvious inequality 4b − 3 < β 2 imply that (a, b) ∈ (2, 60), (3,270), (4,728), (5,1530) .…”

“…K. Ono [20] dealt with several infinite families of such systems and deduced the lack of non-trivial solutions by simply computing the number of representations of certain integers by pairs of suitable ternary quadratic forms. N. Tzanakis, in a very well written exposition [27], advocates the use of linear forms in elliptic logarithms. The same paper contains an ample bibliography, with pointers to other works based on this idea.…”

Integral points and arithmetic progressions on Huff curves

“…The elliptic logarithm method for determining all integer points on an elliptic curve has been applied to a variety of elliptic equations (see e.g. [35,36,38,39,40]). The disadvantage of this approach is that there is no known algorithm to determine the rank of the so-called Mordell-Weil group of an elliptic curve, which is necessary to determine all integral points on the curve.…”