Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> denote nth Fibonacci number. In this paper, we show that the equation <Equation ID="Equ22"> <EquationSource Format="TEX">$\begin{aligned} F_{n}-F_{m}=F_{t}^{a} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo>-</mo> <msub> <mi>F</mi> <mi>m</mi> </msub> <mo>=</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> </mrow> <mi>a</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>has no solution in the integers n, m, t, a satisfying the conditions <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$6\le t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>6</mn> <mo>≤</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$2\le a\le t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$3\le n-m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>. This paper generalizes some previous results.

On the equation $F_{n}-F_{m}=F_{t}^{a}$

期刊论文
发表日期：

2025年8月18日

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Nurettin Irmak,
László Szalay

摘要

Let $F_{n}$ $F_{n}$ denote nth Fibonacci number. In this paper, we show that the equation $\begin{aligned} F_{n}-F_{m}=F_{t}^{a} \end{aligned}$ $\begin{matrix} F_{n} - F_{m} = F_{t}^{a} \end{matrix}$ has no solution in the integers n, m, t, a satisfying the conditions $6\le t$ $6 \leq t$ , $2\le a\le t$ $2 \leq a \leq t$ , and $3\le n-m$ $3 \leq n - m$ . This paper generalizes some previous results.

On the equation \(F_{n}-F_{m}=F_{t}^{a}\)

摘要