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• # Artykuł - szczegóły

## Colloquium Mathematicum

2013 | 130 | 1 | 27-38

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

EN

### Abstrakty

EN
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.

27-38

wydano
2013

### Twórcy

autor
• Department of Mathematics, Sakarya University, TR54187 Sakarya, Turkey
autor
• Department of Mathematics, Bilecik Şeyh Edebali University, TR11030 Bilecik, Turkey