# How to solve this equation

1. Mar 27, 2012

### medwatt

Hello.
I have encountered an equation of the form a^n + b^n = c^n. I know this looks like Fermat's equation. How can I solve for n. It doesn't matter if n is not an integer.
Thanks

2. Mar 28, 2012

### medwatt

Ok the equation I'm trying to solve is :

$\left[\sqrt{2+\sqrt{3}}\right]^{n}$+ $\left[\sqrt{2-\sqrt{3}}\right]^{n}$ = $2^{n}$
$^{}$

I have arrived to this point:

$\left[\sqrt{2+\sqrt{3}}\right]^{2n}$ + $\left[2*\sqrt{2+\sqrt{3}}]\right]^{n}$ - 1 =0

I think the problem is easy but there is a point I'm missing.

I know that 2 is the answer from simple observation and with the knowledge that with Fermat's type of equations an integer solution can't be more than 2.

Last edited: Mar 28, 2012
3. Mar 28, 2012

### Norwegian

So you want to solve

$\left[\frac{\sqrt{2+\sqrt{3}}}{2}\right]^{n}+ \left[\frac{\sqrt{2-\sqrt{3}}}{2}\right]^{n}= 1$

and you know that n=2 is a solution, and I assume you know that an is decreasing for 0<a<1, so that there can't be more solutions.

4. Mar 28, 2012

### Staff: Mentor

okay id try n=0, then n=1 then n=2... and see which ones or one are true. basically you're trying to find n.

also i think you are on the right track using fermat's last theorem (proved by Andrew Wiles in 1995) as an upper bound for n

5. Mar 28, 2012

### medwatt

I need to show that n=2. The answer 2 in this problem was obvious. What would be the general solution to such equations if n is not an integer (i.2 we cannot guess it by simple observation).

6. Mar 28, 2012

### chiro

Do you need an exact analytic answer or will a numerical approximation suffice?

7. Mar 29, 2012

### coolul007

Does n have to be an integer?

8. Mar 29, 2012

### medwatt

It doesn't matter if n is an integer or not. I just want a way to solve the equation because suppose guess the answer was not so easy, how can one solve it without of course a computer.

9. Mar 29, 2012

### coolul007

It resembles a Binet formula.

10. Mar 29, 2012

### medwatt

Yes I think it looks like Binet's formula for Fibonacci sequence.