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Extended Lagrange's four-square theorem
Lagrange's four-square theorem states that every natural number can be
represented as the sum of four integer squares: .
Ramanujan generalized Lagrange's result by providing, up to equivalence, all
quadratic forms that represent all positive
integers. In this article, we prove the following extension of Lagrange's
theorem: given a prime number and , , ,
, such that for all and for all , then there exists
such that for all and This means that, in ,
any system of orthogonal vectors of norm can be completed to a base. We
conjecture that the result holds for every norm . 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
.Comment: 15 page