Extended Lagrange's four-square theorem

• Published in: Electronic notes in discrete mathematics Vol. 68; pp. 209 - 214
• Main Authors: Lacalle, J., Gatti, L.N.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2018
• Subjects: Lagrange's four-square theorem; orthogonal lattices; p-orthonormal base extension theorem; systems of p-orthonormal vectors; p-orthonormal base extension theorem; Lagrange's four-square theorem; orthogonal lattices; systems of p-orthonormal vectors
• ISSN: 1571-0653; 1571-0653
• DOI: 10.1016/j.endm.2018.06.036

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.