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kingwinner

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__Theorem:__The positive primitive solutions of [tex]x^2 + y^2 = z^2[/tex] with y even are [tex]x = r^2 - s^2, y = 2rs, z = r^2 + s^2[/tex], where r and s are arbitrary integers of opposite parity with r>s>0 and gcd(r,s)=1.

**Using this theorem, find all solutions of the equation [tex]x^2 + y^2 = 2z^2[/tex]**

**(hint: write the equation in the form [tex](x+y)^2 + (x-y)^2 = (2z)^2[/tex])**

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The above theorem characterizes all "PRIMITIVE Pythagorean triples", but what is the statement of the theorem that characterizes ALL "Pythagorean triples" (not necessarily primitive)?

(i) The positive solutions of [tex]x^2 + y^2 = z^2[/tex] with y even are precisely [tex]x = (r^2 - s^2)d, y = (2rs)d, z = (r^2 + s^2)d[/tex], where d is any natural number, r and s are arbitrary integers of opposite parity with r>s>0 and gcd(r,s)=1.

(ii) The positive solutions of [tex]x^2 + y^2 = z^2[/tex] with y even are precisely [tex]x = r^2 - s^2, y = 2rs, z = r^2 + s^2[/tex], where r and s are arbitrary integers with r>s>0.

Which one is correct?

I hope someone can help me out. Thank you!