I think it's called "Fermat's 2 squares theorem".

The proof (the one I studied anyway, in Rings & Factorisation lectures) involves factoristation of Gaussian integers, which form a UFD (unique factorisation domain).

a^2 + b^2 = (a+ib)(a-ib)

Looks like it was Euler not Gauss who published the first prrof.

Here's a link

http://www.google.co.uk/search?q=cache:Hqrp8Bz0kGUC:www.math.rutgers.edu/~useminar/squares.pdf+fermat%27s+2+squares+theorem&hl=en&ie=UTF-8

That link also mentions the analogy for sums of three squares, it says the proof is a lot harder than the one for sums of 4 squares. It also tells you how to work out how many combinations there are:

Jacobi's Two Square Theorem: The number of representations of a positive integer as the sum of two squares is equal to four times the difference of the numbers of divisors congruent to 1 and 3 modulo 4.

"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley

[edited by - Paradigm Shifter on January 31, 2003 9:47:08 AM]

You can find circles with arbitrarily large numbers of points with integer coordinates. I''ll sketch the idea. Imagine you want a circle with at least 5000 points with integer coordinates. Start by selecting 5000 rational numbers. On a plane, draw those numbers on the x axis. Now draw a circle of radius one centered in (0,1). For each point that you draw on the x axis, draw a line that joins it with (0,2). The line will intersect the circle in two points: (0,2) and (a,b). You can verify that a and b are rational numbers. Now select the least common multiple of all the a''s and b''s generated this way. Scale the figure by that factor, and the radius one circle is now a big circle that has at least 5000 points with integer coordinates.

Paradigm Shifter, thank you very much!

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Qui fut tout, et qui ne fut rien
Invader''s Realm