Question

Show that x2 + y2 = x²y² have no integer solution except x=y=0​

Answer 1

Answer:

Show that the equation x2+y2+z2=x2y2 has no integer solution,except x=y=z=0.

Let one of the x,y,z be even number.Let x=2p

x2+y2+z2=x2y2

This gives y2+z2 is also even,which means either y,z are both even or y,z are both odd.

So either x=2p,y=2q,z=2r is the solution or x=2p,y=2q+1,z=2r+1 is the solution.I put x=2p,y=2q+1,z=2r+1 in the equation x2+y2+z2=x2y2 and see that it does not satisfy both sides,so it is not the solution.

When I put x=2p,y=2q,z=2r,i get p2+q2+r2=4p2q2

I am confused here how to prove that p2+q2+r2=4p2q2 does not hold true.

Also i cannot solve when x is odd which is the second case.

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