Composition of Sums of Squares with Integer Coefficients

Abstract

This paper is on the construction of shortest composition formulae of the form

$$ \left( {x_1^2 + \cdots + x_r^2} \right)\left( {y_1^2 + \cdots + y_s^2} \right) = z_1^2 + \cdots + z_n^2 $$

(1.1.1)

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.

Research partially supported by NSF Grant DMS-8903412.