On the Distribution of Squares of Hypercomplex Integers Original Research Article

Let image be a real quadratic algebra of dimension sgreater-or-equal, slanted3 which satisfies the basic relations of hypercomplex systems. For a large positive parameter X, let A(X) denote the number of squares α2 with αset membership, variantimage, α integral, and all s components of α2 lying in the interval [−X, X]. With particular regard to Cayleyʹs octaves, and generalizing former results concerning Gaussian integers by H. Müller and W. G. Nowak, and Hurwitz integral quaternions by the author, we show thatA(X)=cXs/2−dX(s−1)/2+O(X(log X)−1/2+X(s−2)/2δ(X))(X→∞),where c and d are certain positive constants depending on s, and δ(X) is any upper bound of the error term in the divisor problem, e.g. δ(X)=X23/73+var epsilon.