sums of two squares (original) (raw)
Theorem.
This was presented by Leonardo Fibonacci in 1225 (in_Liber quadratorum_), but was known also by Brahmagupta and already by Diophantus of Alexandria (III book of his_Arithmetica_).
The proof of the equation may utilize Gaussian integers
as follows:
| (a2+b2)(c2+d2) | =(a+ib)(a-ib)(c+id)(c-id) |
|---|---|
| =(a+ib)(c+id)(a-ib)(c-id) | |
| =[(ac-bd)+i(ad+bc)][(ac-bd)-i(ad+bc)] | |
| =(ac-bd)2+(ad+bc)2 |
Note 1. The equation (1) is the special case n=2of Lagrange’s identity.
Note 2. Similarly as (1), one can derive the identity
| (a2+b2)(c2+d2)=(ac+bd)2+(ad-bc)2. | (2) |
|---|
Thus in most cases, we can get two different nontrivial sum forms (i.e. without a zero addend) for a given product of two sums of squares. For example, the product
attains the two forms 42+72 and 82+12.