On Binary Quadratic Forms with the Semigroup Property

A quadratic form $f$ is said to have the semigroup property if its values at the points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with the semigroup property. If there is an integer bilinear map $s$ such that $f(s(\mathbf x,\mathbf y))=f(\mathbf x)f(\mathbf y)$ for all vectors $\mathbf x$ and $\mathbf y$ from the integer two-dimensional lattice, then the form $f$ has the semigroup property. We give an explicit integer parameterization of all pairs $(f,s)$ with the property stated above. We do not know any other examples of forms with the semigroup property.