On square classes in generalized Fibonacci sequences

Zafer Şiar, and Refik Keskin. "On square classes in generalized Fibonacci sequences." Acta Arithmetica 174.3 (2016): 277-295. <http://eudml.org/doc/286594>.

@article{ZaferŞiar2016,
abstract = {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.},
author = {Zafer Şiar, Refik Keskin},
journal = {Acta Arithmetica},
keywords = {generalized Fibonacci numbers; generalized Lucas numbers; congruences},
language = {eng},
number = {3},
pages = {277-295},
title = {On square classes in generalized Fibonacci sequences},
url = {http://eudml.org/doc/286594},
volume = {174},
year = {2016},
}

TY - JOUR
AU - Zafer Şiar
AU - Refik Keskin
TI - On square classes in generalized Fibonacci sequences
JO - Acta Arithmetica
PY - 2016
VL - 174
IS - 3
SP - 277
EP - 295
AB - 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.
LA - eng
KW - generalized Fibonacci numbers; generalized Lucas numbers; congruences
UR - http://eudml.org/doc/286594
ER -