Quadratic Diophantine equations in the rational and quadratic fields

Abstract

with integral coefficients from the field of rational numbers or from some quadratic field. The quantity A is defined for convenient reference. We can take ap, O without any loss of generality, by the use (if necessary) of linear transformations of determinant unity (so that the number of integral solutions is not changed). First, suppose that the coefficients of (2) are rational integers. If A is negative, then the graph of (2) is finite in extent, and there is at most a finite number of solutions in integers. If A >0, the graph of (2) is a parabola, an hyperbola, or two straight lines, and we prove the following result.

Locations

• Transactions of the American Mathematical Society - View - PDF