On solutions of some Diophantine equations

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. [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. [2] Keskin, R., Güney, Duman, M., “Positive Integer Solutions of Some Pell Equations”, Palestine Journal of Mathematics, 8: 213-226, (2019).
3. [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. [4] Ribenboim, P., “My Numbers, My Friends: Popular Lectures on Number Theory”, Springer Verlag, 1-50, (2000).