On regular polytope numbers

Abstract

Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized as the polygonal number theorem and the Hilbert-Waring problem. In this paper, we shall generalize Lagrange’s sum of four squares theorem further. To each regular polytope $V$ in a Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem of finding the order $g(V)$ of the set of regular polytope numbers associated to $V$.