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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
The solutions are given by $F_{1}-F_{0}=F_{2}-F_{0}=F_{3}-F_{2}=F_{3}-F_{1}=F_{4}-F_{3}=5^{0}$ and $F_{5}-F_{0}=F_{6}-F_{4}=F_{7}-F_{6}=5.$ Then we give a conjecture that says that if $a\geq 2$ and $p>7$ is prime, then the equation $F_{n}-F_{m}=p^{a}$ has no solutions in nonnegative integers $m,n.$
📌 3 Key Takeaways
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Objective & Hypothesis:In this study, we solve the Diophantine equation in the title in nonnegative integers $m,n,$ and $a$.
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Key Findings & Data:In this study, we solve the Diophantine equation in the title in nonnegative integers $m,n,$ and $a$.
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Conclusion & Impact:The solutions are given by $F_{1}-F_{0}=F_{2}-F_{0}=F_{3}-F_{2}=F_{3}-F_{1}=F_{4}-F_{3}=5^{0}$ and $F_{5}-F_{0}=F_{6}-F_{4}=F_{7}-F_{6}=5.$ Then we give a conjecture that says that if $a\geq 2$ and $p>7$ is prime, then the equation $F_{n}-F_{m}=p^{a}$ has no solutions in nonnegative integers $m,n.$
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Diophantine equation · Fibonacci numbers · linear forms in logarithms
Abstract
In this study, we solve the Diophantine equation in the title in nonnegative integers $m,n,$ and $a$. The solutions are given by $F_{1}-F_{0}=F_{2}-F_{0}=F_{3}-F_{2}=F_{3}-F_{1}=F_{4}-F_{3}=5^{0}$ and $F_{5}-F_{0}=F_{6}-F_{4}=F_{7}-F_{6}=5.$ Then we give a conjecture that says that if $a\geq 2$ and $p>7$ is prime, then the equation $F_{n}-F_{m}=p^{a}$ has no solutions in nonnegative integers $m,n.$
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Cite This Article
ERDUVAN, F., KESKİN, R. (2019). Nonnegative integer solutions of the equation $F_{n}-F_{m}=5^{a}$. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 3, 1115–1123. https://ajindex.com/articles/185548