Quadratic Form made a Perfect Power by a New Composition Theorem on Arbitrary Quadratic Forms

Abstract

This paper deals with the diophantine equation $Q(x_1,\,x_2,\,\ldots,\,x_m)= y^n$, where $m$ and $n$ are arbitrary positive integers and $Q(x_1,\,x_2,\,\ldots,\,x_m)$ is an arbitrary quadratic form in the $m$ variables $x_1,\,x_2,\,\ldots,\,x_m$. While solutions of special cases of this equation have been published earlier, the general equation of this type has not been solved till now. To solve this equation, we first show that, given an arbitrary quadratic form $Q(x_1,\,x_2,\,\ldots,\,x_m)$ in $m$ variables, there exists a {\it composition formula} $Q(u_i)\,Q^2(v_i)=Q(w_i)$ where $u_i$ and $v_i$ ($i= 1,\,2,\,\ldots,\,m$) are arbitrary variables and the $w_i$ ($i=1,\,2,\,\ldots,\,m$) are cubic forms in the variables $u_i$ and $v_i$ ($i=1,\,2,\,\ldots,\,m$). This is a new composition formula, different from the standard composition formulae of the type $Q(u_i)Q(v_i)=Q(w_i)$ which are known for certain classes of quadratic forms. As the equation $Q(x_i)=y^n$ is not always solvable, we prove a theorem giving a necessary and sufficient condition for its solvability. We use the aforementioned composition formula to obtain parametric solutions of the equation $Q(x_i)=y^n$, and also give some numerical examples.