On the exponential Diophantine equation Px
 n
 + P
 
 x
 
 
 n+1 + ⋯ + P
 
 x
 
 
 n+k-1 = Pm

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 $k\ge 3$ .

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On the exponential Diophantine equation P^x _n + P ^x _n+1 + ⋯ + P ^x _n+k-1 = P_m

Abstract: In this paper, we find all the solutions of the title Diophantine equation in positive integers (m, n, k, x), where Pi is the ith term of the Pell sequence.

Cited by 2 publications

References 13 publications

Balancing Numbers as Sum of Same Power of Consecutive Balancing Numbers

Balancing Numbers as Sum of Same Power of Consecutive Balancing Numbers

On the Diophantine Equation

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On the exponential Diophantine equation Px n + P x n+1 + ⋯ + P x n+k-1 = Pm

Abstract: In this paper, we find all the solutions of the title Diophantine equation in positive integers (m, n, k, x), where Pi is the ith term of the Pell sequence.

Cited by 2 publications

References 13 publications

Balancing Numbers as Sum of Same Power of Consecutive Balancing Numbers

Balancing Numbers as Sum of Same Power of Consecutive Balancing Numbers

On the Diophantine Equation

Contact Info

Product

Resources

About

On the exponential Diophantine equation P^x _n + P ^x _n+1 + ⋯ + P ^x _n+k-1 = P_m