On the Diophantine equation \(x^2-kxy+ky^2+ly=0\), \(l\in\{1,2,4,8\}\). (English) Zbl 1290.11059

Karaatlı, Olcay; Ṣiar, Zafer

Afr. Diaspora J. Math. 14, No. 1, 24-29 (2012).

Summary: In this study, we consider the Diophantine equation \(x^{2}-kxy+ky^{2}+ly=0,l\in \{ 1,2,4,8\}\) and determine the values of \(k\) when the equation has infinitely many positive integer solutions \(x\) and \(y\).

MSC:

11D09	Quadratic and bilinear Diophantine equations
11B37	Recurrences
11B39	Fibonacci and Lucas numbers and polynomials and generalizations

Keywords:

quadratic Diophantine equations; Pell equations

References:

[1]	T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York , 1981. · Zbl 0042.26702
[2]	John P. Robertson, Solving the generalized Pell equation \(x^{2}-Dy^{2}=N,\) http://hometown.aol.com/jpr2718/pell.pdf, (Description of LMM Algorithm for solving Pell’s equation), May 2003.
[3]	Michael J. Jacobson and Hugh C. Williams, Solving the Pell Equation, Springer , 2006. · Zbl 1177.11027 · doi:10.1007/978-0-387-84923-2
[4]	H. W. Lenstra Jr, Solving the Pell equation, Notices Amer. Math. Soc. 49 (2002), pp 182-192. · Zbl 1126.11312
[5]	R. A. Mollin, A. J. van der Poorten, and H. C. Williams, Halfway to a solution of \(X^{2}-DY^{2}=-3\) , J. Théor. Nombres Bordeaux , 6 (1994), pp 421-457. · Zbl 0820.11015 · doi:10.5802/jtnb.123
[6]	R. A. Mollin, Quadratic Diophantine equations \(x^{2}-dy^{2}=c^{n}\) , Irish Math. Soc. Bull. 58 (2006), pp 55-68. · Zbl 1144.11025
[7]	P. Stevanhagen, A density conjecture for negative Pell equation, pp 187-200, Computational Algebra and Number Theory, Sydney, 1992, Kluwer Acd. Publ., Dordrecht. · Zbl 0838.11066 · doi:10.1007/978-94-017-1108-1_13
[8]	R. Keskin, O. Karaatlı, and Z. Şiar, On the Diophantine equation \(x^{2}-kxy+y^{2}+2^{n}=0\) , Miskolc Mathematical Notes, · Zbl 1274.11079
[9]	R. Keskin and B. Demirtürk, Solutions of Some Diophantine Equations Using Generalized Fibonacci and Lucas Sequences Ars Combinatoria, · Zbl 1313.11055
[10]	J. H. Cohn, Twelve Diophantine Equations, Arch. Math. 65 (1995), pp 130-133. · Zbl 0837.11013 · doi:10.1007/BF01270690
[11]	James P. Jones, Representation of Solutions of Pell Equations Using Lucas Sequences, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae , 30 (2003), pp 75-86. · Zbl 1047.11017
[12]	A. Marlewski and P. Zarzycki, Infinitely Many Solutions of the Diophantine Equation \(x^{2}-kxy+y^{2}+x=0\) , Computers and Mathematics with Applications , 47 (2004), pp 115-121. · Zbl 1053.11024 · doi:10.1016/S0898-1221(04)90010-7
[13]	P. Yuan and Y. Hu, On the Diophantine Equation \( x^{2}-kxy+y^{2}+lx=0\) , \(l\in \left\{ 1,2,4\right\} \), Computers and Mathematics with Applications , 61 (2011), pp 573-577. · Zbl 1217.11031