On solutions of some Diophantine equations

Year 2025, Volume: 38 Issue: 2, 891 - 897, 01.06.2025

Refik Keskin , Kübra Ged , Fatih Erduvan

https://doi.org/10.35378/gujs.1378010

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.

Keywords

Diophantine equations , Generalized Fibonacci numbers , Generalized Lucas numbers

References

• [1] Keskin, R., “Solutions of some quadratic Diophantine equations”, Computer and Mathematics with Applications, 60(8): 2225-2230, (2010). DOI: 10.1016/j.camwa.2010.08.012
• [2] Keskin, R., Güney, Duman, M., “Positive Integer Solutions of Some Pell Equations”, Palestine Journal of Mathematics, 8: 213-226, (2019).
• [3] Keskin, R., Güney, Duman, M., “Positive Integer Solutions of Some Second-Order Diophantine equations”, Annale Ştiintifice ale Universitatii, “Alexandru Ioan Cuza”, din Iaşi Matematica, (New Series), 3: 649-664, (2017).
• [4] Ribenboim, P., “My Numbers, My Friends: Popular Lectures on Number Theory”, Springer Verlag, 1-50, (2000).