2014/02/21 by Raza, Zahid, Malik, Hafsa Masood
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1402.5206
Let a,b,c be any positive integers such that c| ab and di^± is a square free positive integer of the form di^±=a2k b2l± i cm where k,l ≥ m and i=1,2. The main focus of this paper to find the fundamental solution of the equation x2-di^± y2=1 with the help of the continued fraction of √(di^±). We also obtain all the positive solutions of the equations x2-di^± y2=± 1 and x2-di^± y2=± 4 by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form Fdi^±(x, y) = x2-di^± y2 including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation F_Δdi^± (x, y) = 1 in terms of di^±. We generalized all the results of the papers [2], [9], [26], and [37].