Curious properties of generalized Lucas numbers
Autor: | Hayder R. Hashim |
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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 |
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