For sums of squares formulas of the form (x1 + · · · + xr)(y 1 + · · · + y s) = z 1 + · · · + z t where the zi are bilinear functions of the xi and yi, we have two wellknown lower bounds on the size of t given r and s. One was obtained independently by Hopf and Stiefel, and another by Atyiah. These bounds are given by requiring certain binomial coefficients be divisible by certain powers of 2. ...