Solutions of polynomial Pell's equation

Abstract

Let D = F 2 + 2 G be a monic quartic polynomial in Z [ x ] , where deg G < deg F . Then for F / G ∈ Q [ x ] , a necessary and sufficient condition for the solution of the polynomial Pell's equation X 2 − D Y 2 = 1 in Z [ x ] has been shown. Also, the polynomial Pell's equation X 2 − D Y 2 = 1 has nontrivial solutions X , Y ∈ Q [ x ] if and only if the values of period of the continued fraction of D are 2, 4, 6, 8, 10, 14, 18, and 22 has been shown. In this paper, for the period of the continued fraction of D is 4, we show that the polynomial Pell's equation has no nontrivial solutions X , Y ∈ Z [ x ] .