Solutions of equations x2−(p2q2±3p)y2=±kt

Abstract

In the present paper, we have solved the equation $x^2-(p^2{q}^2\pm 3p)y^2=k^t,x^2-(p^2{q}^2\pm 5p)y^2=k^t$ and expressed its positive integer solutions in terms of generalized Fibonacci, generalized Lucas and generalized Pell, generalized Pell–Lucas sequences. With the help of this equation, we have found units of $Z\left[\sqrt{885}\right]$ and $Z\left[\sqrt{915}\right]$ in terms of generalized Fibonacci, generalized Lucas, generalized Pell and generalized Pell–Lucas numbers.