We study the problem of finding all integer solutions of the Diophantine equations x2 − 5Fnxy − 5 (−1) y2 = ±Ln, x2 − Lnxy + (−1) y2 = ±5F 2 n , and x2 − Lnxy + (−1) y2 = ±F 2 n . Using these equations, we also explore all integer solutions of some other Diophantine equations.

Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of associated coefficient ring, we can produce sequences these equations. It known that be written as tuples linear recurrence sequences. We show for certain families forms defined over quartic fields, there exist integrally equivalent making any one fixed coordinate sequence divisibility sequen...

A new algorithm for finding the minimal solutions of systems of linear Diophantine equations has recently been published. In its description the emphasis was put on the mathematical aspects of the algorithm. In complement to that, in this paper another presentation of the algorithm is given which may be of use for anyone wanting to implement it.

S. Lang [L] conjectured in 1974 that a hyperbolic algebraic variety defined over a number field has only finitely many rational points, and its analogue over function fields. We discuss the Nevanlinna-Cartan theory over function fields of arbitrary dimension and apply it for Diophantine property of hyperbolic projective hypersurfaces (homogeneous Diophantine equations) constructed by Masuda-Nog...

In this paper we construct sets of type (d1, d2) in the projective Hjelmslev plane. For computational purposes we restrict ourself to planes over Zps with p a prime and s > 1, but the method is described over general Galois rings. The existence of sets of type (d1, d2) is equivalent to the existence of a solution of a Diophantine system of linear equations. To construct these sets we prescribe ...