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I'm practicing proofs and I'm stuck. Here it is:

Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x^2 + y^2 = z^2. Evidently I'm supposed to start by setting x, y, and z like this:

x = m^2 - n^2

y = 2mn

z = m^2 + n^2

So then we have:

(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2

m^4 + n ^4 - 2(mn)^2 + 4(mn)^2 = m ^4 + n^4 +2(mn)^2

m ^4 + n^4 +2(mn)^2 = m ^4 + n^4 +2(mn)^2

Now I'm sort of at a standstill. I understand that I can plug any integer into m and n and x^2 + y^2 = z^2 will be true, but I'm not sure how to prove it.

Also, another way to show the proof would be to let x be:

x = 3m

y = 4m

z = 5m

Since any number can be plugged into m then there are infinite solutions.

However, I would like to understand how to derive the proof from my first method.

I'm really trying to understand this stuff so any help would be appreciated.

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# Proof of infinitely many solutions

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