Extended Lagrange's four-square theorem

Abstract

We prove the following extension of Lagrange's theorem: given a prime number p and v1,…,vk∈Z4,1≤k≤3, such that ∥vi∥2=p for all 1≤i≤k and 〈vi|vj〉=0 for all 1≤i<j≤k, then there exists v=(x1,x2,x3,x4)∈Z4 such that 〈vi|v〉=0 for all 1≤i≤k and∥v∥=x12+x22+x32+x42=p This means that, in Z4, any system of orthogonal vectors of norm p can be completed to a base. We conjecture that the result holds for every norm p≥1.