On the discriminant of a quadratic field with intermediate fractions of negative norm and the decomposability of its representing polynomial

Abstract

The work is devoted to the study of Diophantine equation $x^2-y^2(p^{2}-4q)=4t$, where $p=l+u(k^2-1)(l(k^2-1)-2k)$, $q=u(lk^3-2k^2-kl+1)+km+1$, $l=k+m(k^{2}-1)$, numbers k,m,u are nonnegative integers, number k is odd, and the right hand side 4t of the equation is sufficiently small positive integer. We give a complete description of solutions of the Diophantine equation.