The polynomial Pell's equation is X2 - DY2 = 1, where D is a polynomial with integer coefficients and the solutions X, Y must be polynomials with integer coefficients. Let D = A2 + 2C be a polynomial in L[x], where deg C<deg A. Then for pB = pA/C ∈ L[x], 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.
ASJC Scopus subject areas
- Algebra and Number Theory