Simultaneous integer sequences and solving the Pell equation
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Let k ≥ 1 be a fixed integer. In this work, we set two simultaneous integer sequences defined by Xn = (4k2 + 1)Xn-1 + (4k2 + 1)Xn-2 - Xn-3 and Yn = (4k2 + 1)Yn-1 + (4k2 + 1)Yn-2 - Yn-3 for n ≥ 3 with initial terms X0 = 1, X1 = 2k2+1, X2 = 8k4+8k2+1 and Y0 = 0, Y1 = 2k, Y2 = 8k3+4k and derived some algebraic identities on them. Further, we are able to determine all integer solutions of the Pell equation x2-(k2 +1)y2 = 1 as (xn, yn) = (Xn, Yn) for every n ≥ 1.
Abstract
Let k ≥ 1 be a fixed integer. In this work, we set two simultaneous integer sequences defined by Xn = (4k2 + 1)Xn-1 + (4k2 + 1)Xn-2 - Xn-3 and Yn = (4k2 + 1)Yn-1 + (4k2 + 1)Yn-2 - Yn-3 for n ≥ 3 with initial terms X0 = 1, X1 = 2k2+1, X2 = 8k4+8k2+1 and Y0 = 0, Y1 = 2k, Y2 = 8k3+4k and derived some algebraic identities on them. Further, we are able to determine all integer solutions of the Pell equation x2-(k2 +1)y2 = 1 as (xn, yn) = (Xn, Yn) for every n ≥ 1.
참고문헌 (10)
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