A challenging Number Theory problem

[jay_shark]

Show that if A and B can be written as the sum of the squares of 4 integers , then so can their product A*B .

I found this question to be very difficult .

[bigpooch]

First of all, Lagrange proved that every non-negative
integer can be written as a sum of four squares (and since
the product of two non-negative integers is a non-negative
integer, it's clear).

What you are asking for is probably Euler's four-square
identity; here's a link to Wikipedia:

http://en.wikipedia.org/wiki/Euler%27s_four-square_identity

[jay_shark]

I didn't know that this was attributed to Euler , but thx for the link .

[Enrique]

Fermat conjectured that every number would be the sum of n n-agonal numbers.
Gauss proved for triangular (1796), Lagrange for squares (1770) and Cauchy proved the whole thing in 1813.

Awesome theorem.

[bigpooch]

I've read about the result (first time was in Apostol's
book on analytic number theory), but haven't looked at the
proof.

A very nice theorem indeed!