Characterisation of Quadratic Integral Pellian Polynomials

A polynomial D∈ℤ[x] is called Pellian over ℤ if the polynomial Pell equation P^2-DQ^2=1 has a non-trivial solution in ℤ[x] . It is an open problem to determine all the polynomials D∈ℤ[x] that are Pellian over ℤ . In the literature, the study of the Pellian polynomials over ℤ has been mostly restricted to monic polynomials as there are many key difficulties involved in the non-monic case. In this article, we overcome those difficulties and characterise all the quadratic polynomials in ℤ[x] that are Pellian over ℤ by providing a necessary and sufficient condition for Pellianity over ℤ . This seems to be the first instance of a comprehensive study on the Pellianity over ℤ of non-monic quadratic polynomials in ℤ[x] . A key difficulty is to find solutions in ℤ[x] , when the necessary condition is satisfied. Our proof exhibits a constructive method to do so.