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ON THE DIOPHANTINE EQUATION 
$(P^{(k)}_n)^2+(P^{(k)}_{n+1})^2=P^{(k)}_m$
Published online by Cambridge University Press: 06 March 2024
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 
$k\ge 3$.
MSC classification
Primary:
11D61: Exponential equations
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
This research was supported by ADA University Faculty Research and Development Funds.
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