Extended Lagrange's four-square theorem

Abstract

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$ quadratic forms $ax_1^2+bx_2^2+cx_3^2+dx_4^2$ that represent all positive integers. In this article, we prove the following extension of Lagrange's theorem: given a prime number $p$ and $v_1\in Z^4$, $\dots$, $v_k\in Z^4$, $1\leq k\leq 3$, such that $\|v_i\|^2=p$ for all $1\leq i\leq k$ and $\langle v_i|v_j\rangle=0$ for all $1\leq i<j\leq k$, then there exists $v=(x_1,x_2,x_3,x_4)\in Z^4$ such that $\langle v_i|v\rangle=0$ for all $1\leq i\leq k$ and $$ \|v\|^2=x_1^2+x_2^2+x_3^2+x_4^2=p $$ This means that, in $Z^4$, any system of orthogonal vectors of norm $p$ can be completed to a base. We conjecture that the result holds for every norm $p\geq 1$. The problem comes up from the study of a discrete quantum computing model in which the qubits have Gaussian integers as coordinates, except for a normalization factor $\sqrt{2^{-k}}$.