Re: Sum of squares

chrizm7@xxxxxxxxx writes:

If a^2 + b^2 = x^2 + y^2 (and a, b, x, y all not zero)

then I am 99% sure either a = x, b = y or a = y, b = x.

It isn't true.

Geometrically, of course, a^2 + b^2 = r is the equation of a circle.
So if x^2 + y^2 = r as well, both equations represent the same circle.

And there are lots of different points on a circle. There may even
be lots of integer points on a circle.

But I would like a symbolic proof without appealing to geometry.

a^2 - x^2 = y^2 - b^2

(a + x) (a - x) = (y + b) (y - b)

To find an example in positive integers, take any positive integer that
can be factored in two different ways as the product of two distinct
positive integers that are both odd or both even. For example,
15 * 1 = 5 * 3
a + x = 15, a - x = 1 => a = 8, x = 7
y + b = 5, y + b = 3 => y = 4, b = 1
8^2 + 1^2 = 7^2 + 4^2
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.