A note on the diophantine equation
Let , , , be positive integers such that , , is even and is odd. In this paper we prove that if and either or is an odd prime power, then the equation has only the positive integer solution with .
To each indefinite integral binary quadratic form , we may associate the geodesic in through the roots of quadratic equation . In this paper we study the asymptotic distribution (as discriminant tends to infinity) of the angles between these geodesics and one fixed vertical geodesic which intersects all of them.
Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation has only the positive integer solution (x,y,z) = (1,1,2).