Fundamental Solutions of _u ² _-5v ² _{= -4r} ²

Abstract

In this paper we consider the number of fundamental solution possessed by the Pell equation

$$ {u^2} - 5{v^2} = - 4{r^2} $$

((1))

where r is a given positive integer. In [2], it is shown that (1) has only the fundamental solutions $\pm r + r\sqrt 5$ and $4r + 2\sqrt 5$ provided r has no prime factor p with p≡ ± 1 (mod 10). Moreover, it is conjectured that, if m ∣r where $m = \prod _i^t = 1P_i^bi$ and $Pi \equiv \pm 1$(mod 10) for each i, then the number of fundamental solutions of (1) is given by

$$ s(m) = 3\tau ({m^2}) $$

((2))

where r (m ²) is the number of divisors of m ². Here we prove that the conjecture is correct.