Arithmetic progressions on Pell equations
The authors consider the solutions \((X,Y)\) of the Pell equations \(X^2-dY^2=m\) and the possible arithmetic progression for the \(X\)-coordinates or the \(Y\)-coordinates. They prove several finiteness results. We choose a typical theorem among these results. Theorem. Let \(m\) be a fixed nonzero integer. There there are only finitely many positive integres \(d\) such that there are three solutions \((X_1,Y_1)\), \((X_2,Y_2)\) and \((X_3,Y_3)\) to the Pell equation \(X^2-dY^2=m\), such that \(X_1<X_2<X_3\) or \(Y_1<Y_2<Y_3\) is an arithmetical progression, except the trivial cases with \(X_2Y_2=0\) and \(X_3=-X_1\) or \(Y_3=-Y_1\). Moreover if such an \(X\)-progression (respectively, a \(Y\)-progression) exists then \(d\leq 3| m| \), (respectively, \(d\leq 9 m^2\)).
scientific article; zbMATH DE number 5285721