The mathematician, Fermat, said that any prime of the form can be expressed as a sum of two squares. However, it turns out amazingly, that this representation is alsounique. Accept by faith the existence of Fermat's theorem and prove uniqueness.

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- August 26th 2007, 02:50 PMThePerfectHackerProblem 35
The mathematician, Fermat, said that any prime of the form can be expressed as a sum of two squares. However, it turns out amazingly, that this representation is also

**unique**. Accept by faith the existence of Fermat's theorem and prove uniqueness. - September 3rd 2007, 05:10 AMSkyWatcher
I think I have got a solution, but's its also prooving the existence of the two numbers : no acceptation by faith!

would it fit?

(i hope not, because i feel lazy and though it's not very long it's a little tricky because it's involving the distinction betwenn numbers and their 'correspondant' modulo 4k+1=p'):o - September 4th 2007, 04:48 PMThePerfectHacker
Say that .

Note that .

Thus,

.

Our goal now is to show either . Now follows at once from the equation . But if we will show this implies that .

Now if then . Thus, .

Now we have established that .

Say the first one is true . Then . But since . Say therefore that . Substitute into we get . But then .

Now using a similar argument with we find that .

I hope you like this proof. I really find it beautiful how divisibility arguments lead us to this conclusion. - June 17th 2009, 12:50 PMBruno J.
Suppose .

Then

But is a unique factorization domain hence the primes are the same up to unit factors and rearrangement (i.e. : . This yields that and we are done.