The polynomial Pell's equation is X 2 À DY 2 ¼ 1; where D is a polynomial with integer coefficients and the solutions X ; Y must be polynomials with integer coefficients. Let D ¼ A 2 þ 2C be a polynomial in Z½x; where deg Codeg A: Then for pB ¼ pA=CAZ½x; p a prime, a necessary and sufficient condition for which the polynomial Pell's equation has a nontrivial solution is obtained. Furthermore, all solutions to the polynomial Pell's equation satisfying the above condition are determined. r 2004 Elsevier Inc. All rights reserved.

No abstract

, a necessary and sufficient condition for the solution of the polynomial Pell's equationhas been shown. Also, the polynomial Pell's equation X 2 − DY 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].

Let N(n) be the set of all integers that can be written in the form where ∊(n) → 0 as n → ∞, answering a question of P. Erdös and R. L. Graham.

No abstract