Pfister’s Theorem on Sums of Squares

A similar 4-square identity was discovered by Euler in 1748: (x1 + x 2 2 + x 2 3 + x 2 4)(y 2 1 + y 2 2 + y 2 3 + y 2 4) = (x1y1 − x2y2 − x3y3 − x4y4) + (x1y2 + x2y1 + x3y4 − x4y3) + (x1y3 − x2y4 + x3y1 + x4y2) + (x1y4 + x2y3 − x3y2 + x4y1). This was rediscovered by Hamilton (1843) in his work on quaternions. Soon thereafter, Graves (1843) and Cayley (1845) independently found an 8-square identity: the product (x1 + · · ·+ x8)(y 1 + · · ·+ y2 8) equals (x1y1 − x2y2 − x3y3 − x4y4 − x5y5 − x6y6 − x7y7 − x8y8) + (x1y2 + x2y1 + x3y4 − x4y3 + x5y6 − x6y5 − x7y8 + x8y7) + (x1y3 − x2y4 + x3y1 + x4y2 + x5y7 + x6y8 − x7y5 − x8y6) + (x1y4 + x2y3 − x3y2 + x4y1 + x5y8 − x6y7 + x7y6 − x8y5) + (x1y5 − x2y6 − x3y7 − x4y8 + x5y1 + x6y2 + x7y3 + x8y4) + (x1y6 + x2y5 − x3y8 + x4y7 − x5y2 + x6y1 − x7y4 + x8y3) + (x1y7 + x2y8 + x3y5 − x4y6 − x5y3 + x6y4 + x7y1 − x8y2) + (x1y8 − x2y7 + x3y6 + x4y5 − x5y4 − x6y3 + x7y2 + x8y1). This formula had been discovered about 35 years earlier, by Degen, but that was unknown to Hamilton, Cayley, and Graves. Mathematicians began searching next for a 16-square identity but results were inconclusive for a long time. The general question we ask is: for which n is there any identity