Curious properties of generalized Lucas numbers

Autor: Hayder R. Hashim
Rok vydání: 2021
Předmět:
Zdroj: Boletín de la Sociedad Matemática Mexicana. 27
ISSN: 2296-4495
1405-213X
DOI: 10.1007/s40590-021-00391-7
Popis: Let $$\{V_n(P,Q)\}$$ be the Lucas sequence of the second kind at the nonzero relatively prime parameters P and Q. In this paper, we present techniques for studying the solutions (x, n) with $$x \ge 2, n \ge 0$$ of any Diophantine equation of the form $$\begin{aligned} \frac{1}{V_n(P_2,Q_2)}=\sum _{k=1}^{\infty }\frac{V_{k-1}(P_1,Q_1)}{x^k} \end{aligned}$$ in both cases $$(P_1,Q_1)=(P_2,Q_2)$$ and $$(P_1,Q_1)\ne (P_2,Q_2)$$ , where $$Q_1, Q_2 \in \{-1,1\}$$ . Furthermore, we represent the procedures of these techniques in case of $$ -2 \le P_1, P_2 \le 4$$ .
Databáze: OpenAIRE