A description of continued fraction expansions of quadratic surds represented by polynomials

Abstract

The principal thrust of this investigation is to provide families of quadratic polynomials fD k ðX Þ ¼ f 2 k X 2 þ 2e k X þ Cg kAN ; where e 2 k À f 2 k C ¼ n (for any given nonzero integer n) satisfying the property that for any X AN; the period length c k ¼ cð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi D k ðX Þ p Þ of the simple continued fraction expansion of ffiffiffiffiffiffiffiffiffiffiffiffiffiffi D k ðX Þ p is constant for fixed k and lim k-N c k ¼ N: This generalizes, and completes, numerous results in the literature, where the primary focus was upon jnj ¼ 1; including the work of this author, and coauthors, in Mollin (Far East

Solutions of polynomial Pell's equation

“…Solving Pell's equation in Z [x] has been studied by Mollin [2][3][4][5][6], Nathanson [7], Ramasamy [8], Webb and Yokota [9,10,12]. The authors [9] Given D = F 2 + 2G with deg G < deg F , it is known [1,12] …”