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We establish partition regularity of the generalised Pythagorean equation infive or more variables. Furthermore, we show how Rado's characterisation of apartition 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 thuscompletely describe which diagonal forms are partition regular and which arenot, given sufficiently many variables. In addition, we prove a supersaturatedversion of Rado's theorem for a linear equation restricted either to squaresminus one or to logarithmically-smooth numbers.

Rado's criterion over squares and higher powers