productofsumsofsquares

The product of sums of squares is a mathematical identity that relates the product of two sums of two squares to another sum of two squares. Specifically, it states that the product of two numbers, each of which is a sum of two squares, can itself be expressed as a sum of two squares.

The identity can be written as:

(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2

and also as

(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2

This identity is named after the French mathematician Jacques Philippe Marie Binet, who published it in 1816, though it was known to Fermat earlier. It is a special case of the Brahmagupta–Fibonacci identity, which generalizes this to the product of sums of four squares.

The product of sums of squares has applications in number theory, particularly in the study of numbers that can be expressed as the sum of two squares. For example, if a prime number p can be written as the sum of two squares, and another prime number q can be written as the sum of two squares, then their product pq can also be written as the sum of two squares. This property extends to any integer that is a product of primes, each of which is a sum of two squares.