Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to solve this equation

  1. Mar 27, 2012 #1
    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. jcsd
  3. Mar 28, 2012 #2
    Ok the equation I'm trying to solve is :

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

    I have arrived to this point:

    [itex]\left[\sqrt{2+\sqrt{3}}\right]^{2n}[/itex] + [itex]\left[2*\sqrt{2+\sqrt{3}}]\right]^{n}[/itex] - 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
  4. Mar 28, 2012 #3
    So you want to solve

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

    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.
     
  5. Mar 28, 2012 #4

    jedishrfu

    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
     
  6. Mar 28, 2012 #5
    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).
     
  7. Mar 28, 2012 #6

    chiro

    User Avatar
    Science Advisor

    Do you need an exact analytic answer or will a numerical approximation suffice?
     
  8. Mar 29, 2012 #7
    Does n have to be an integer?
     
  9. Mar 29, 2012 #8
    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.
     
  10. Mar 29, 2012 #9
    It resembles a Binet formula.
     
  11. Mar 29, 2012 #10
    Yes I think it looks like Binet's formula for Fibonacci sequence.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How to solve this equation
Loading...