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The purpose of this article is to give the solutions of the inverse problemfor Pellian equations. For any rational number $0< a/b < 1$, the fundamentaldiscriminants $D$ satisfying $(\lfloor \sqrt{D} \rfloor b + a)^2 - D b^2 = 4$are given in terms of a quadratic progression. There were studies about thisproblem based on symmetric sequences $\{a_1,\...,a_{l-1}\}$ and periodiccontinued fractions $[a_0,\bar{a_1,\...,a_{l-1},a_l}]$, but in this article wesolve the problem in a completely different way with simpler parameters. Theresult is obtained by measuring the quality of approximation of a rationalnumber to $\sqrt{d}$ or $\frac{1+\sqrt{d}}{2}$, and by defining a shortinterval attached to each rational number. On this formulation we also showthat for almost all square-free integer $d$, $d$ is the least element of theprescribed quadratic progression for some $a/b$.

Inverse problem for Pell equation and real quadratic fields of the least type