On solutions of polynomial Pell's equations and the continued fraction factorization algorithm

研究多项式佩尔方程首先做了阿贝尔在连接找到一个椭圆积分，可以表示使用一个初等函数。这是在有理数领域中进行的，当我们把多项式Pell方程的解限制为整系数多项式时，已知的结果只适用于一元四次多项式。我们在2003年证明了多项式Pell方程对D = A^2+2C且A/C∈Q[x]有非平凡整系数多项式解的一个充分必要条件<D>。我们还从小周期的角度研究了多项式Pell方程，对于一元四次多项式D，我们能够证明不存在周期为3的连分式展开式，对于一元多项式D，我们能够证明连分式展开式的周期值是偶数当且仅当多项式Pell方程X^2-DY^2 =对于一元四次多项式D，我们证明了Pell方程X^2-DY^2 = 1在Q[x]中有非平凡解的充要条件是连分式展开式的周期为2，4，6，8，10，14，18，22.

Study on the polynomial Pell's equation was first done by Abel in the connection of finding an elliptic integral which can be expressed using an elementary function. This was done in the field of rational numbers.When we restrict solutions of the polynomial Pell's equation to be integer coefficient polynomial, the known result is only for a monic quartic polynomial. We have shown in 2003, a necessary and sufficient condition for the polynomial Pell's equation has a nontrivial integer coefficient polynomial solution for D = A^2+2C and A/C∈Q[x].In this research, collaborating with Prof.Webb, we have studied the polynomial Pell's equation using the period of continued fraction expansions of √<D> in the connection with rational points on the elliptic curve arising from the partial quotients. We also have studied the polynomial Pell's equation by looking at the small periods.For D a monic quartic polynomial, we are able to show that there is no period 3 continued fraction expansion.For D a monic polynomial, we are able to show that the values of period of continued fraction expansions are even if and only if the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution.For D a monic quartic polynomial, we are able to show that the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution in Q[x] if and only if the values of the period of continued fraction expansions are 2,4,6,8,10,14,18,22.