Question

if a^2 + b^2 has a factor x, then why is it true that we can write a=mx +/- c, b=nx +/- d, where c and d are at most half of x in absolute value ?

I'm taking this from here http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares

I'm trying to follow along on the proof and I'm up to part 4 of Euler's proof, where it says the thing I described above. Thanks...

Answer 1

I see now. It isn't very obvious why that is true.

Answer 2

Basically, They're looking at the number modulo x. However, if \(a\pmod x\) is greater than one half of x, just subtract x so that c is between \[-\frac{x}{2} \leq c\leq \frac{x}{2}\]Every number can be expressed this way, and that's what the proof relies on.

Answer 3

Does this make it clearer to you @scarydoor ?

Answer 4

Hmn. I was just thinking about this and I came to realise that all they're saying is that, given a number a and a number x, both integers, you can express a as an integer multiple of x plus or minus some other integer, so that this other integer is at least less than half the value of x. So, you just take multiples of x and keep going until a is within half of one multiple from the result.

I was confused because I was looking at a^2+b^2 and thinking that had something to do with the derivation, but it doesn't.

I think you are right in saying that it's about mod values, since they're multiples of x, although I didn't think of it that way.

Thanks for your help!