## On the Diophantine equation $x^2-dy^4=1$ with prime discriminant II

### Volume 105 / 2006

Colloquium Mathematicum 105 (2006), 51-55
MSC: 11D41, 11B39.
DOI: 10.4064/cm105-1-6

#### Abstract

Let $p$ denote a prime number. P. Samuel recently solved the problem of determining all squares in the linear recurrence sequence $\{ T_n \}$, where $T_n$ and $U_n$ satisfy $T_n^2-pU_n^2=1$. Samuel left open the problem of determining all squares in the sequence $\{ U_n \}$. This problem was recently solved by the authors. In the present paper, we extend our previous joint work by completely solving the equation $U_n=bx^2$, where $b$ is a fixed positive squarefree integer. This result also extends previous work of the second author.