Saturday, 12 March 2016

number theory - Prove that there exist infinitely many primitive Pythagorean triples x,y,z whose even member x is a perfect square.

Prove that there exist infinitely many primitive Pythagorean triples x,y,z whose even member x is a perfect square. [Hint: consider the triple 4n2,n44,n4+4, where n is an arbitraty odd integer.]



What I got:



Using the hint. 4n2,n44,n4+4 is a Phytagorean triple if x2+y2=z2. Replacing and solving the equation it is clear that, (4n2)2+(n44)2=(n4+4)2 where n is odd is indeed a Pythagorean triple with x=4n2=(2n2)2 a perfect square.




Now I have to prove that gcd(4n2,n44,n4+4)=1.



But I'm stuck here. I tried gcd(n44,n4+4)=1 without success. Any ideas?

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