The k–generalized Fibonacci sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(F_m^{(k)})_{m\ge 2-k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> <mo>-</mo> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the linear recurrent sequence of order k whose first k terms are <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0, \ldots , 0, 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and each term afterwards is the sum of the preceding k terms. The case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> corresponds to the well known Fibonacci sequence. In Gómez and Luca (Lith. Math. J. 56(4):503–517, 2016), the multiplicative independence between terms of the same k-generalized Fibonacci sequence was studied. In this paper, we find all the multiplicative dependent pairs <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(F_m^{(k)},u_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is a Fibonacci, a Lucas or a Pell number.

Multiplicative dependence of k-Fibonacci numbers with the Fibonacci, Lucas, and Pell sequences

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2026年1月31日

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Carlos Gómez,
Jhonny C. Gómez,
Florian Luca

摘要

The k–generalized Fibonacci sequence $(F_m^{(k)})_{m\ge 2-k}$ ${(F_{m}^{(k)})}_{m \geq 2 - k}$ is the linear recurrent sequence of order k whose first k terms are $0, \ldots , 0, 1$ $0, \dots, 0, 1$ and each term afterwards is the sum of the preceding k terms. The case $k=2$ $k = 2$ corresponds to the well known Fibonacci sequence. In Gómez and Luca (Lith. Math. J. 56(4):503–517, 2016), the multiplicative independence between terms of the same k-generalized Fibonacci sequence was studied. In this paper, we find all the multiplicative dependent pairs $(F_m^{(k)},u_n)$ $(F_{m}^{(k)}, u_{n})$ where $u_n$ $u_{n}$ is a Fibonacci, a Lucas or a Pell number.