Pell方程x^2-Dy^2=±2的解的递推性质

The Recurrence Relations of the Solutions of Pell Equation x^2-Dy^2=±2

在各类不定方程中,Pell方程x2-Dy2=N是一类基础而重要的Diophantine方程,其正整数解与实二次域的基本单位以及其它代数数论理论有密切联系,对解高次丢翻图方程以及有关递推数列问题有广泛且深入的应用.利用Pell方程的基本解的性质,对方程x2-Dy2=±2的通解进行了讨论,获得了该方程解的一个三阶递推性质,证明了文献(A.Tekcan.Irish.Math.Soc.Bulletin,2004,54(1):73-89.)提出的一个猜想.最后,提出了关于Pell方程x2-Dy2=-2可解性的一些待解决的问题.

Among all kinds of Diophantine equations, Pell equation x^2 -Dy^2 = N is basic and important. Its positive integer solution is closely linked with the fundamental unit of real quadratic field and other algebraic number theory and is extensively applied to solve high-order Diophantine equations and the problems on recursive number serieses. Using the properties of the fundamental solution of Pell equation, the equation x^2 - Dy^2 = ± 2 is discussed, and a third-order recursive property of the solution is obtained, a conjecture of A. Tekcan（ Irish. Math. Soc. Bulletin,2004,54（ 1 ） ：73 - 89. ） is proved. Finally, a number of usolved problems of Pell equation x^2 -Dy^2 = -2 are put forward.