The Generalized Fibonacci and Lucas Solutions of The Pell Equations x²-(a²b²-b)y²=N and x²-(a²b²-2b)y²=N

2013/03/07 by Peker, Bilge, Senay, Hasan
#11A55 #11B39 #11B50 #11B99 #11D09 #11D45 #11D79 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1303.1838

Abstract

In this study, we find continued fraction expansion of sqrt(d) when d=a²b²-b and d=a²b²-2b where a and b are positive integers. We consider the integer solutions of the Pell equations x²-(a²b²-b)y²=N and x²-(a²b²-2b)y²=N when N is +-1,+-4. We formulate the n-th solution (x_n,y_n) by using the continued fraction expansion. We also formulate the n-th solution (x_n,y_n) in terms of generalized Fibonacci and Lucas sequences.

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