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 $a{x}_1^2+b{x}_2^2+c{x}_3^2+d{x}_4^2$ that represent all positive integers. In this article, we prove the following extension of Lagrange's theorem: given a prime number $p$ and $v_1\in Z^4$, $\dots$, $v_k\in Z^4$, $1\le k\le 3$, such that $\Vert v_i{\Vert }^2=p$ for all $1\le i\le k$ and $⟨v_i|v_j⟩=0$ for all $1\le i<j\le k$, then there exists $v=(x_1,x_2,x_3,x_4)\in Z^4$ such that $⟨v_i|v⟩=0$ for all $1\le i\le k$ and $$\Vert v{\Vert }^2=x_1^2+x_2^2+x_3^2+x_4^2=p$$ 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 $p\ge 1$. 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 $\sqrt{2^{-k}}$.

15 pages