Quadratic form made a perfect power by a new composition theorem on arbitrary quadratic forms (Q648496)

This paper deals with the diophantine equation of the form \((\star)\;Q(x_{1},\dots,x_{m})=y^{n}\), where \(m, n\) are positive integers and \(Q\) is quadratic form in variables \(x_{1},\dots,x_{m}\). The problem of finding integer solutions of this equation is investigated with the help of a new composition formula for \(Q\). This formula states that if \(U=(u_{1},\dots, u_{m})\) and \(V=(v_{1},\dots,v_{m})\) are vectors of variables then there are cubic forms \(W_{i}=W_{i}(U,V)\) for \(i=1,\dots,m\), such that \(Q(U)Q(V)^2=Q(W_{1},\dots,W_{m})\). In particular, the author proves that if \(n\) is odd then the equation (\(\star)\) always has a non-zero solution in integers. Moreover, if \(n\) is even then (\(\star)\) has solution in integers if and only if \(Q(x_{1},\dots,x_{m})=y^{2}\) has a solution in integers.