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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
In this note, we find all positive integer solutions to the Diophantine equations x^2-kxy-y^2=±k, x^2-(k^2+4) y^2=±4k, x^2-2pxy-y^2=±2p, x^2-kxy-y^2=±k(k^2+4) in terms of generalized Fibonacci and Lucas numbers.
📌 3 Key Takeaways
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Objective & Hypothesis:Let k and p be prime numbers with p>2.
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Key Findings & Data:Let k and p be prime numbers with p>2.
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Conclusion & Impact:In this note, we find all positive integer solutions to the Diophantine equations x^2-kxy-y^2=±k, x^2-(k^2+4) y^2=±4k, x^2-2pxy-y^2=±2p, x^2-kxy-y^2=±k(k^2+4) in terms of generalized Fibonacci and Lucas numbers.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Diophantine equations · Generalized Fibonacci numbers · Generalized Lucas numbers
Abstract
Let k and p be prime numbers with p>2. In this note, we find all positive integer solutions to the Diophantine equations x^2-kxy-y^2=±k, x^2-(k^2+4) y^2=±4k, x^2-2pxy-y^2=±2p, x^2-kxy-y^2=±k(k^2+4) in terms of generalized Fibonacci and Lucas numbers.
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Cite This Article
Keskin, R., Ged, K., Erduvan, F. (2025). On solutions of some Diophantine equations. Gazi University Journal of Science, Volume: 38 Issue: 2. https://doi.org/10.35378/gujs.1378010