Theorem: The positive primitive solutions of with y even are , 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
(hint: write the equation in the form )
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I am not sure how to do this problem. I don't even understand how to get the equation in the hint.
I hope someone can help me out. Thank you!
[also under discussion in math links forum]
m and n have to have the same parity, is it because of the last equation (2z must be even)???
But the theorem above requires r and s to be integers of opposite parity with r>s>0 and (r,s)=1. Do we need m and n to satisfy all of these requirements as well? (I think we need to ensure all these conditions are satsified for m and n, otherwise the theorem above won't even be applicable?)
Thanks for helping!
The theorem above is for PRIMITIVE Pyth. triples. Is there another theorem that says x,y,z is a Pythagorean triple (may or may not be primitive, just ANY Pythagorean triple) IF AND ONLY IF x,y,z are of the form where r>s>0 are integers??
Also, in our problem, when we solve for x,y,z, we get terms like and , but are they well-defined? (i.e. are they necessarily integers?)
Theorem: The positive primitive solutions of with y even are precisely , where r and s are arbitrary integers of opposite parity with r>s>0 and gcd(r,s)=1.
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 with y even are precisely , 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 with y even are precisely , where r and s are arbitrary integers with r>s>0.
Which one is correct??