engleski

Arithmetic progressions and Pellian equations

We consider arithmetic progressions on Pellian equations x^2 - d y^2 = 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 innitely many pairs d ; m parametrized by points of an elliptic curve of positive rank for which the corresponding Pellian equations have solutions whose y-component form a six-term arithmetic progression. Then we exhibit many six-term progressions whose elements are the y-components of solutions for an equation of the form x^2 - d y^2 = m with small coefficients d, m and also several particular seven-term examples. Finally we show a procedure for searching five-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 recent results of Dujella, Petho and Tadic, and Petho and Ziegler.

Pellian equations ; arithmetic progression

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