NOTES ON GENERALISED INTEGRAL POLYNOMIAL PELL EQUATIONS

Abstract Given a nonzero integer n , Gupta and Saha [‘Integer solutions of the generalised polynomial Pell equations and their finiteness: the quadratic case’, Canad. Math. Bull. , to appear] classified all polynomials $x^2+ax+b\in {\mathbb {Z}}[x]$ for which the polynomial Pell equation $P^2-(x^2+ax+b)Q^2=n$ has solutions ${P,Q\in {\mathbb {Z}}[x]}$ with $Q\neq 0$ . We generalise their work to the equation $P^2-(f^2+af+b)Q^2=nR$ , where f is a fixed polynomial in ${\mathbb {Z}}[x]$ . As an application of our results, we study the equation $P^2-D(f)Q^2=n$ , where D is a monic, quartic and non square-free polynomial in ${\mathbb {Z}}[x]$ . This extends Theorem 1.4 of Scherr and Thompson [‘Quartic integral polynomial Pell equations’, J. Number Theory 259 (2024), 38–56].