The history of the 2-, 4- and 8-square identities

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 1995.

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 1919b.

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 (1919a) showed that this result still holds if is replaced by any other field K, provided char K ≠ 2.

¹⁰ See, for example, Rajwade 1993.