- #1

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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

- Thread starter medwatt
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- #1

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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

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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.

[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:

- #3

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[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 a

- #4

jedishrfu

Mentor

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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

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- #6

chiro

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

- #7

coolul007

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Does n have to be an integer?

- #8

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- #9

coolul007

Gold Member

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It resembles a Binet formula.

- #10

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

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