Thread: Diophantine eq

Thanks heaps but there is another one which I cant figure out. I have to prove that for each positive integer n there exist integers x,y,z > 1 so that:
x^2+y^2=17z^(4n)

:S

How does induction work with 4 variables?

No need for induction. Try to guess a solution which will be valid for all (and dependent on ), using a single solution to .

(Sorry if this hint was too big)

I think AdamC and I are in the same class ;-)

The question asks us specifically to prove the proposition using simple induction. But, Unbeatable0, if I've taken your hint correctly, the values for x, y & z need to be greater than 1.

Originally Posted by AdamC

Thanks heaps but there is another one which I cant figure out. I have to prove that for each positive integer n there exist integers x,y,z > 1 so that:
x^2+y^2=17z^(4n)

:S

How does induction work with 4 variables?

Why do you worry about the variables? Worry about n...

For ...check

Suppose for , and we shall prove for n+1. But

, so using the inductive hypothesis...

Tonio

Originally Posted by tonio

Why do you worry about the variables? Worry about n...

For ...check

Here, you should worry about the variables, because the question specifies integers x,y,z > 1. (But it's not hard to adapt the base case example to satisfy those conditions.)

Originally Posted by Opalg

Here, you should worry about the variables, because the question specifies integers x,y,z > 1. (But it's not hard to adapt the base case example to satisfy those conditions.)

Good point. Completely missed that bigger than 1 thingy.

Tonio