On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ

Refik Keskin ; Zafer Şiar

Colloquium Mathematicae, Tome 131 (2013), p. 27-38 / Harvested from The Polish Digital Mathematics Library

Résumé

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and $U_{n+1} = PUₙ - QU_{n-1}$ for n ≥ 1, and V₀ = 2, V₁ = P and $V_{n+1} = PVₙ - QV_{n-1}$ for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and $V_{r} ≠ 1$. We show that there is no integer x such that $Vₙ = V_{r}Vₘx²$ when m ≥ 1 and r is an even integer. Also we completely solve the equation $Vₙ = VₘV_{r}x²$ for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and Q ≡ 1 (mod 4), the equation $Vₙ = VₘV_{r}x²$ has no solutions for m ≥ 1 and r ≥ 1. Moreover, we show that when P > 1 and Q = ±1, there is no generalized Lucas number Vₙ such that $Vₙ = VₘV_{r}$ for m > 1 and r > 1. Lastly, we show that there is no generalized Fibonacci number Uₙ such that $Uₙ = UₘU_{r}$ for Q = ±1 and 1 < r < m.

Détail

Publié le : 2013-01-01

Zbl 1272.11031

EUDML-ID : urn:eudml:doc:283993