Arithmetic Progressions and Pellian Equations

We consider arithmetic progressions on Pellian equations x2 − d y2 = m, i.e. integral solutions such that the y-coordinates are in arithmetic progression. We construct explicit infinite families of d,m for which there exists a five-term arithmetic progression in the y-coordinate, and we prove the existence of infinitely many pairs d,m parametrized by the points of an elliptic curve of positive rank for which the corresponding Pellian equations have six-term progressions as solutions.Then we exhibit many six-term progressions with the y-components being solutions for an equation of the form x2 − d y2 = m with small coefficients d,m and also several particular seventerm examples. Finally we show a procedure for searching 5-term arithmetic progressions for which there exist a couple of pairs (d1,m1) and (d2,m2) for which the progression is a solution of the associated Pellian equations. These results extend and complement the results in recent papers by Dujella, Pethő and Tadić and Pethő and Ziegler.