# Thread: Questions about Fermat numbers?

1. ## Questions about Fermat numbers?

I have recently happened upon an interesting problem that I cannot solve. Here it is:
a. Factorise $\displaystyle x^3+1$.
b. Factorise $\displaystyle x^n+1$ when $\displaystyle {n}\geq{3}$ is odd. Use this to prove the following:
i. If $\displaystyle 2^n+1$ is prime, then $\displaystyle n$ must be a power of $\displaystyle 2$.
ii. Show that $\displaystyle 2^3^2+1$ is not prime.

I can't seem to make any headway on the problem, in particular the very last part. All help is appreciated on this problem.

2. For (a),(b) show that $\displaystyle x^n+1 = (x+1)(x^{n-1}-x^{n-2}+\cdots +x^2-x+1)$, as n is odd.

i) Use (b) to show that if $\displaystyle n=2^kq$, where q is odd, then

the factorisation of $\displaystyle x^n+1$ is obtained by substituting $\displaystyle x=x^{2^k}$ into $\displaystyle (x^q+1)$

So this means if n is not a power of 2....?

ii) You can show that $\displaystyle 2^{32}+1$ is divisable by 5.

Can you do modulo-calculations?

3. Originally Posted by Dinkydoe For (a),(b) show that $\displaystyle x^n+1 = (x+1)(x^{n-1}-x^{n-2}+\cdots +x^2-x+1)$, as n is odd.

i) Use (b) to show that if $\displaystyle n=2^kq$, where q is odd, then

the factorisation of $\displaystyle x^n+1$ is obtained by substituting $\displaystyle x=x^{2^k}$ into $\displaystyle (x^q+1)$

So this means if n is not a power of 2....?

ii) You can show that $\displaystyle 2^{32}+1$ is divisable by 5.

Can you do modulo-calculations?
I cannot do modulo-calculations, however I don't think that $\displaystyle 2^3^2+1$ is divisible by $\displaystyle 5$.

4. Originally Posted by JRichardson1729 I cannot do modulo-calculations, however I don't think that $\displaystyle 2^3^2+1$ is divisible by $\displaystyle 5$.
$\displaystyle 2^3^2+1$ was proven composite by Euler. Explanation in this PDF

http://www.maa.org/editorial/euler/H...oring%20F5.pdf

5. Originally Posted by undefined $\displaystyle 2^3^2+1$ was proven composite by Euler. Explanation in this PDF

http://www.maa.org/editorial/euler/H...oring%20F5.pdf
I knew of this, however I don't see how it uses the factorisation of $\displaystyle x^n+1$ when $\displaystyle {n}\geq{3}$ and is odd as the question asks for.

6. Originally Posted by JRichardson1729 I knew of this, however I don't see how it uses the factorisation of $\displaystyle x^n+1$ when $\displaystyle {n}\geq{3}$ and is odd as the question asks for.
Since 32 is not odd, the property does not directly apply. Maybe it is an error in the problem statement.

7. Sorry, made a lame error in (ii)....but indeed. I agree with undefined. I don't see any relation between (ii) and the previous.

8. I found this video, YouTube - the 5th Fermat number is not a prime (german), that shows this, but, unfortunately, I don't think that it uses the factorization of $\displaystyle x^n+1$ when $\displaystyle {n}\geq{3}$ and is odd and I don't know how to adapt it so that it does (if that is possible at all).

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