Abstract
Abstract
A generalisation of the well-known Pell sequence
$\{P_n\}_{n\ge 0}$
given by
$P_0=0$
,
$P_1=1$
and
$P_{n+2}=2P_{n+1}+P_n$
for all
$n\ge 0$
is the k-generalised Pell sequence
$\{P^{(k)}_n\}_{n\ge -(k-2)}$
whose first k terms are
$0,\ldots ,0,1$
and each term afterwards is given by the linear recurrence
$P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+\cdots +P^{(k)}_{n-k}$
. For the Pell sequence, the formula
$P^2_n+P^2_{n+1}=P_{2n+1}$
holds for all
$n\ge 0$
. In this paper, we prove that the Diophantine equation
$$ \begin{align*} (P^{(k)}_n)^2+(P^{(k)}_{n+1})^2=P^{(k)}_m \end{align*} $$
has no solution in positive integers
$k, m$
and n with
$n>1$
and
$k\ge 3$
.
Publisher
Cambridge University Press (CUP)