We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x2 − dy2 = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic
progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations
of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.