  We establish partition regularity of the generalised Pythagorean equation in
five or more variables. Furthermore, we show how Rado's characterisation of a
partition regular equation remains valid over the set of positive $k$th powers,
provided the equation has at least $(1+o(1))k\log k$ variables. We thus
completely describe which diagonal forms are partition regular and which are
not, given sufficiently many variables. In addition, we prove a supersaturated
version of Rado's theorem for a linear equation restricted either to squares
minus one or to logarithmically-smooth numbers.
