In this paper, we find all the solutions of the Diophantine equation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F_{k,1}^2+2F_{k,2}^2+\cdots +mF_{k,m}^2=F_{k,n}^q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>F</mi> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>F</mi> <mrow> <mi>k</mi> <mo>,</mo> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>m</mi> <msubsup> <mi>F</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> <mi>q</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> in positive integer variables (m, n), where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F_{k,i}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$i^{th}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>i</mi> <mrow> <mi mathvariant="italic">th</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> term of the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k^{th}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>k</mi> <mrow> <mi mathvariant="italic">th</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> Fibonacci sequence, k a positive integer and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$q\in \{1,2\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>.

On the Diophantine equation $\sum _{j=1}^{m}jF_{k,j}^2=F_{k,n}^q$

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2026年1月24日

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Euloge Tchammou,
Alain Togbé

摘要

In this paper, we find all the solutions of the Diophantine equation $F_{k,1}^2+2F_{k,2}^2+\cdots +mF_{k,m}^2=F_{k,n}^q$ $F_{k, 1}^{2} + 2 F_{k, 2}^{2} + \dots + m F_{k, m}^{2} = F_{k, n}^{q}$ in positive integer variables (m, n), where $F_{k,i}$ $F_{k, i}$ is the $i^{th}$ $i^{th}$ term of the $k^{th}$ $k^{th}$ Fibonacci sequence, k a positive integer and $q\in \{1,2\}$ $q \in {1, 2}$ .

On the Diophantine equation \(\sum _{j=1}^{m}jF_{k,j}^2=F_{k,n}^q\)

摘要