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Math Help - Questions about Fermat numbers?

  1. #1
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    Questions about Fermat numbers?

    I have recently happened upon an interesting problem that I cannot solve. Here it is:
    a. Factorise x^3+1.
    b. Factorise x^n+1 when {n}\geq{3} is odd. Use this to prove the following:
    i. If 2^n+1 is prime, then n must be a power of 2.
    ii. Show that 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.
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  2. #2
    Senior Member Dinkydoe's Avatar
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    For (a),(b) show that 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 n=2^kq, where q is odd, then

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

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

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

    Can you do modulo-calculations?
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  3. #3
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    Quote Originally Posted by Dinkydoe View Post
    For (a),(b) show that 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 n=2^kq, where q is odd, then

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

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

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

    Can you do modulo-calculations?
    I cannot do modulo-calculations, however I don't think that 2^3^2+1 is divisible by 5.
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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by JRichardson1729 View Post
    I cannot do modulo-calculations, however I don't think that 2^3^2+1 is divisible by 5.
    2^3^2+1 was proven composite by Euler. Explanation in this PDF

    http://www.maa.org/editorial/euler/H...oring%20F5.pdf
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  5. #5
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    Quote Originally Posted by undefined View Post
    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 x^n+1 when {n}\geq{3} and is odd as the question asks for.
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  6. #6
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by JRichardson1729 View Post
    I knew of this, however I don't see how it uses the factorisation of x^n+1 when {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.
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  7. #7
    Senior Member Dinkydoe's Avatar
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    Sorry, made a lame error in (ii)....but indeed. I agree with undefined. I don't see any relation between (ii) and the previous.
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  8. #8
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    I found this video, , that shows this, but, unfortunately, I don't think that it uses the factorization of x^n+1 when {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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