Composition of Sums of Squares with Integer Coefficients

This paper is on the construction of shortest composition formulae of the form 1.1.1 $$ \left( {x_1^2 + \cdots + x_r^2} \right)\left( {y_1^2 + \cdots + y_s^2} \right) = z_1^2 + \cdots + z_n^2 $$ in which z 1 ,..., z n are polynomials in x 1 ,..., x r , y 1 ,..., y s with integer coefficients. We shall call an identity of this form an [r, s,n]ℤ formula. Composition formulae are generalizations of the classical 2–, 4–, 8–square identities which express the multiplicative property of the norms of complex numbers, quaternions and octonians respectively. The impossibility of a 16–square identity, viz. a [16,16,16] ℤ formula, was discovered in the late 1840’s. This suggested the problem of determining, for given r and s, the smallest integer n, denoted r *ℤ s, for which there exists an [r, s, n] ℤ formula. The purpose of this paper is to determine the precise values of r *ℤ s in the range 10 ≤ r, s ≤ 16.