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

100%

Keskin R., Şiar Z.

Colloquium Mathematicum

2013

tom 130

nr 1

27-38

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.

On square classes in generalized Fibonacci sequences

100%

Şiar Z., Keskin R.

Acta Arithmetica

2016

tom 174

nr 3

277-295

Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and $U_{n+1} = PUₙ + QU_{n-1}$, $V_{n+1} = PVₙ + QV_{n-1}$ for n ≥ 1. In this paper, when w ∈ {1,2,3,6}, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x².

Ograniczanie wyników

1 Acta Arithmetica

1 Colloquium Mathematicum

2 Keskin R.

2 Şiar Z.

1 2016

1 2013

Wyniki wyszukiwania

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

On square classes in generalized Fibonacci sequences