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,n4−4,n4+4, where n is an arbitraty odd integer.]
What I got:
Using the hint. 4n2,n4−4,n4+4 is a Phytagorean triple if x2+y2=z2. Replacing and solving the equation it is clear that, (4n2)2+(n4−4)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,n4−4,n4+4)=1.
But I'm stuck here. I tried gcd(n4−4,n4+4)=1 without success. Any ideas?
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