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GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2 AND 7kx2

  • Received : 2014.06.05
  • Published : 2015.09.30

Abstract

Generalized Fibonacci and Lucas sequences ($U_n$) and ($V_n$) are defined by the recurrence relations $U_{n+1}=PU_n+QU_{n-1}$ and $V_{n+1}=PV_n+QV_{n-1}$, $n{\geq}1$, with initial conditions $U_0=0$, $U_1=1$ and $V_0=2$, $V_1=P$. This paper deals with Fibonacci and Lucas numbers of the form $U_n$(P, Q) and $V_n$(P, Q) with the special consideration that $P{\geq}3$ is odd and Q = -1. Under these consideration, we solve the equations $V_n=5kx^2$, $V_n=7kx^2$, $V_n=5kx^2{\pm}1$, and $V_n=7kx^2{\pm}1$ when $k{\mid}P$ with k > 1. Moreover, we solve the equations $V_n=5x^2{\pm}1$ and $V_n=7x^2{\pm}1$.

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  2. On the equation V n = w x 2 ∓ 1 vol.23, pp.2, 2017, https://doi.org/10.1016/j.ajmsc.2016.06.004
  3. Generalized Fibonacci numbers of the form $$wx^{2}+1$$ w x 2 + 1 vol.73, pp.2, 2016, https://doi.org/10.1007/s10998-016-0133-4