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

Volume 105 / 2006

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


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.


  • D. PoulakisDepartment of Mathematics
    Aristotle University of Thessaloniki
    University Campus
    541 24 Thessaloniki, Greece
  • P. G. WalshDepartment of Mathematics
    University of Ottawa
    585 King Edward St.
    Ottawa, Ontario, Canada, K1N 6N5

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image