For an integer <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(P^{(k)}_{n})_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be the k-generalized Pell sequence whose first k terms are <InlineEquation ID="IEq3"> <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 given by the kth order linear recurrence <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$P^{(k)}_{n}=2P^{(k)}_{n-1}+P^{(k)}_{n-2} +\cdots +P^{(k)}_{n-k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. For the usual Pell sequence <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(P_n)_n=(P^{(2)}_{n})_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> <mo>=</mo> <msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> the formula <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$P_{n}^2+P_{n+1}^2=P_{2n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> holds for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we are interested in finding when the sum of x-th powers of two consecutive generalized Pell numbers is again a generalized Pell number. This paper continues and extends a previous work of Rihane et al. (Turk J Math 43:1640–1649, 2019) who considered this problem with Pell numbers, and the recent work of Hasanalizade (Bull Aust Math Soc p. 5, 2024) who solved the particular case <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

An Exponential Diophantine Equation with Generalized Pell Numbers

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2025年10月22日

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Jhon J. Bravo,
Florian Luca,
Fabian Pomeo

摘要

For an integer $k\ge 2$ $k \geq 2$ , let $(P^{(k)}_{n})_n$ ${(P_{n}^{(k)})}_{n}$ be the k-generalized Pell sequence whose first k terms are $0,\ldots , 0, 1$ $0, \dots, 0, 1$ and each term afterwards is given by the kth order linear recurrence $P^{(k)}_{n}=2P^{(k)}_{n-1}+P^{(k)}_{n-2} +\cdots +P^{(k)}_{n-k}$ $P_{n}^{(k)} = 2 P_{n - 1}^{(k)} + P_{n - 2}^{(k)} + \dots + P_{n - k}^{(k)}$ . For the usual Pell sequence $(P_n)_n=(P^{(2)}_{n})_n$ ${(P_{n})}_{n} = {(P_{n}^{(2)})}_{n}$ the formula $P_{n}^2+P_{n+1}^2=P_{2n+1}$ $P_{n}^{2} + P_{n + 1}^{2} = P_{2 n + 1}$ holds for all $n\ge 0$ $n \geq 0$ . In this paper, we are interested in finding when the sum of x-th powers of two consecutive generalized Pell numbers is again a generalized Pell number. This paper continues and extends a previous work of Rihane et al. (Turk J Math 43:1640–1649, 2019) who considered this problem with Pell numbers, and the recent work of Hasanalizade (Bull Aust Math Soc p. 5, 2024) who solved the particular case $x=2$ $x = 2$ .