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

Refik Keskin, and Zafer Şiar. "On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ." Colloquium Mathematicae 130.1 (2013): 27-38. <http://eudml.org/doc/283993>.

@article{RefikKeskin2013,
abstract = {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.},
author = {Refik Keskin, Zafer Şiar},
journal = {Colloquium Mathematicae},
keywords = {Lucas sequence; congruence},
language = {eng},
number = {1},
pages = {27-38},
title = {On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ},
url = {http://eudml.org/doc/283993},
volume = {130},
year = {2013},
}

TY - JOUR
AU - Refik Keskin
AU - Zafer Şiar
TI - On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ
JO - Colloquium Mathematicae
PY - 2013
VL - 130
IS - 1
SP - 27
EP - 38
AB - 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.
LA - eng
KW - Lucas sequence; congruence
UR - http://eudml.org/doc/283993
ER -