Re: Sum of squares-
- From: JEMebius <jemebius@xxxxxxxxx>
- Date: Sat, 19 May 2007 17:00:49 +0100
chrizm7@xxxxxxxxx wrote:
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. But what
makes this true, because it is clearly not true if we remove the
squares:
2 + 5 = 1 + 6
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.
But I would like a symbolic proof without appealing to geometry.
The subject of "Sums of Squares" is a mathematical universe in its own right.
Let me just recall just the two theorems that are most relevant to your observation.
Both are applicable to positive integers.
(A)
A prime number p = 4n+1 is the sum of two squares in an essentially unique way.
(B)
A number which is the product of N different prime numbers of the form 4n+1 is the sum of two squares in essentially 2^(N-1) different ways.
Therefore your conjecture is true (even 100% instead of 99%) only in the case that a^2 + b^2 is a prime number.
Literature:
Hardy and Wright: An introduction to the theory of numbers. Oxford, 1938; reprinted at least four times...
and many more books! But this one is among the best, perhaps the best introduction to number theory.
Happy studies: Johan E. Mebius
.
- References:
- Sum of squares
- From: chrizm7
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