For an integer <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_806_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <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"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_806_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{n}^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> be the k-generalized Pell sequence which starts with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_806_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <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> (k terms) and each term afterwards is given by the linear recurrence <Equation ID="Equ42"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_806_Article_Equ42.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="359" /> </MediaObject> <EquationSource Format="TEX">$ P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}, \quad \text {for all }n \ge 2. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <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> <mo>,</mo> <mspace width="1em" /> <mtext>for all</mtext> <mspace width="0.333333em" /> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </Equation>In this paper, our study focuses on Fermat and Mersenne numbers and we determine all of them, which are expressed as products of two k-Pell numbers.

Fermat or Mersenne numbers as products of two k-Pell numbers

期刊论文
发表日期：

2025年6月25日

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Safia Seffah,
Salah Eddine Rihane,
Alain Togbé

摘要

For an integer $k\ge 2$ $k \geq 2$ , let $P_{n}^{(k)}$ $P_{n}^{(k)}$ be the k-generalized Pell sequence which starts with $0,\ldots ,0,1$ $0, \dots, 0, 1$ (k terms) and each term afterwards is given by the linear recurrence $ P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}, \quad \text {for all }n \ge 2. $ $P_{n}^{(k)} = 2 P_{n - 1}^{(k)} + P_{n - 2}^{(k)} + \dots + P_{n - k}^{(k)}, for all n \geq 2 .$ In this paper, our study focuses on Fermat and Mersenne numbers and we determine all of them, which are expressed as products of two k-Pell numbers.