Abstract
The identity for the product of two sums of two squares has been known since ancient times. There also exist versions of this identity for 4 and 8 squares, these dating from the eighteenth and nineteenth centuries respectively. In this article I survey the history of such identities and explain why those for 2, 4 and 8 squares are the only ones.
Notes
1 For these results, see, for example, Jackson Citation1995.
2 It should be noted that Diophantus admitted not just integral squares but also rational squares. The consideration of purely integral squares comes from Fermat.
3 My source for Diophantus' Arithmetica is the 1965 edition of Sir Thomas Heath's original 1885 translation, the first translation of Arithmetica into English.
4 Given a positive integer n, we factor out the largest square which divides it and write n=s 2m so that no square divides m; m is called the square-free part of n. Let us take 54 as an example: we write 54=32×6. Clearly, 6 is not divisible by a square so 6 is the square-free part of 54.
5 IV.29, IV.30, and V.14, where the integers in question are 13, 5 and 30, respectively.
6 Namely, quaternions and octonions. For a brief introduction to quaternions and octonions in the context of the n-square identities, see Dickson Citation1919b.
7 And, of course, x 2 y 2=(xy)2, the 1-square identity!
8 Note that this use of the letter ‘N’ is slightly different from Gauss' use in the alternative form of the 4-square identity.
9 Leonard Dickson (Citation1919a) showed that this result still holds if is replaced by any other field K, provided char K ≠ 2.
10 See, for example, Rajwade Citation1993.