Prove that if (2 to the power n) - 15 = x square, then n = 4 or n = 6.

Thanks very much guys:)

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- Dec 3rd 2006, 07:36 PMsuedenationnumber theory, solve diophantine equation
Prove that if (2 to the power n) - 15 = x square, then n = 4 or n = 6.

Thanks very much guys:) - Dec 3rd 2006, 07:47 PMThePerfectHacker
We need,

If,

Then,

Thus,

Thus,

The factorization of 15 is,

Examine each of the cases to get the result.

Note,

Thus, we have only two possibilities,

The equations respectively,

In each case we have,

----

If,

then,

--->Impossible.

If and

Then,

Thus,

This is a Pellian look-a-like equation.

(Now I am hoping that it has no solution to it).

I have to think about this diophantine equation.

Basically I need to show,

Has no solutions. - Dec 3rd 2006, 07:58 PMThePerfectHacker
Aha! Solved it.

The diophantine equation,

Has no solution!

The left hand side needs to be divisible by 15

Thus,

This means the Legendre symbols have the value,

But that is not true!

Because by Euler's criterion,

- Dec 3rd 2006, 08:39 PMsuedenation
Thanks sooooo much, you are a genius....hehe:D