Arithmetic progressions on Pell equations

Abstract

In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions (X, Y ) whose X-coordinates or Y -coordinates are in arithmetic progression.

On geometric progressions on Pell equations and Lucas sequence

“…Similarly as in [8] we obtain as a corollary that for small m there are no three term geometric progressions, in particular we find a method to determine for fixed m all d such that (2) provides geometric progressions in their solution set. Corollary 1.…”

“…However, to find d, m ∈ Z such that a given geometric progression is admitted by (2) is not easy, even if we allow negative d. In view of [8,Theorem 5 and Theorem 7] we show:…”

“…However in this paper we consider equation (2) for all d, m ∈ Z. Some years ago Pethő and Ziegler [8] considered the vertical problem for this case, i.e. they considered arithmetic progressions on such Diophantine equations and obtained effective results.…”

“…, 9 22 3 1, 7, 49 28 2 2, 6, 18 28 8 1, 3, 9 33 3 1, 8, 64 34 2 1, 55, 3025 37 21 2, 22, 242 37 84 1, 11, 121 41 2 8, 20, 50 41 8 4, 10, 25 56 11 2, 10, 50 56 44 1, 5, 25 57 7 1, 4, 16 57 87 1, 8, 64 63 2 3, 9, 27 63 18 1, 3, 9 65 It is very surprising that the following theorem on linear relations on the solution set of Pell equations contains as a corollary an upper bound for three term arithmetic progressions (cf. [8,Theorem 1]) as well as an upper bound for three term geometric progressions (Theorem 1).…”

Some applications of the abc-conjecture to the diophantine equation $qy^m=f(x)$

“…(ii) We have ν X (u) ≪ ln Xd(u) for sufficiently large X (see [13,Lemma 3] for a more precise estimation). Therefore we may proceed as in (i): …”