Diophantine equations and identities

The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are i ) x 2 − m y 2 = ± 1 i i ) x 3 + m y 3 + m 2 z 3 − 3 m x y z = 1 i i i ) Some fifth degree diopantine equations Infinitely many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before. It is known that the solutions of Pell's equation are well exploited. We include it here because we shall use a common method to solve these three above mentioned equations and the method becomes very simple in Pell's equations case. Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields.