On the Number of Perfect Binary Quadratic Forms

Abstract

A perfect form is a form {\small $f=mx^2+ny^2+kxy$} with integral coefficients {\small $(m,n,k)$} such that {\small $f(\mathbb{Z}^2)$} is a multiplicative semigroup. The growth rate of the number of perfect forms in cubes of increasing side {\small $L$} in the space of the coefficients is known for small cubes, where all perfect forms are known. A form is perfect if its coefficients belong to the image of a map, {\small $Q$}, from {\small $\mathbb{Z}^4$} to {\small $\mathbb{Z}^3$}. This property of perfect forms allows us to estimate from below the growth rate of their number for larger values of {\small $L$}. The conjecture that all perfect forms are generated by {\small $Q$} allows us to reformulate results and conjectures on the numbers of the images {\small $Q(\mathbb{Z}^4)$} in cubes of side {\small $L$} in terms of the numbers of perfect forms. In particular, the proportion of perfect elliptic forms in a ball of radius {\small $R$} should decrease faster than {\small $R^{-3/4}$} and the proportion of all perfect forms in a ball of radius {\small $R$} should decrease faster than {\small $2/\sqrt{R}$}.