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Math Help - Not a power of 2

  1. #1
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    Not a power of 2

    Show that k^n+k^{n-1}+\ldots +k+1 can't be a power of 2 if k>1 ( n,k\in\mathbb{Z^+}).
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  2. #2
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    Quote Originally Posted by james_bond View Post
    Show that k^n+k^{n-1}+\ldots +k+1 can't be a power of 2 if k>1 ( n,k\in\mathbb{Z^+}).

    This is false for n = 1 and k = 3 or 7, or 15 or 31 or....

    Tonio
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  3. #3
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    the claim would be true if we also assume that n > 1. the proof is quite easy: let f_n(k)=k^n + k^{n-1} + \cdots + k + 1, \ \ k > 1, \ n>1. it's clear that if n is even, then f_n(k) is odd.

    so we may assume that n is odd. the proof now is by induction over n. we have f_3(k)=k^3+k^2+k+1=(k+1)(k^2+1). suppose f_3(k) is a power of 2. then k=2r+1, for

    some r \geq 1, and so k^2+1=4r(r+1)+2, which is never a power of 2 because r > 0. if n=2m+1 > 3, then the identity f_{2m+1}(k)=f_m(k) (k^{m+1} + 1) completes the proof.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    the claim would be true if we also assume that n > 1. the proof is quite easy: let f_n(k)=k^n + k^{n-1} + \cdots + k + 1, \ \ k > 1, \ n>1. it's clear that if n is even, then f_n(k) is odd.

    so we may assume that n is odd. the proof now is by induction over n. we have f_3(k)=k^3+k^2+k+1=(k+1)(k^2+1). suppose f_3(k) is a power of 2. then k=2r+1, for

    some r \geq 1, and so k^2+1=4r(r+1)+2, which is never a power of 2 because r > 0. if n=2m+1 > 3, then the identity f_{2m+1}(k)=f_m(k) (k^{m+1} + 1) completes the proof.
    Not to sound like a creeper, but I notice how long it takes you to write yoru responses to post. You take very long to write up your proofs. Is that because you are being careful?
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    Quote Originally Posted by Drexel28 View Post
    Not to sound like a creeper, but I notice how long it takes you to write yoru responses to post. You take very long to write up your proofs. Is that because you are being careful?
    ok, i guess i misunderstood your question. i get it now. yes, i like my response to be as accurate as possible. also sometimes while i'm replying to a question i find a better solution and so i start

    my response all over again.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    ok, i guess i misunderstood your question. i get it now. yes, i like my response to be as accurate as possible. also sometimes while i'm replying to a question i find a better solution and so i start

    my response all over again.
    Haha, it's cool NCA. I just always wondered why it took you so long. There were a couple times you replied to some of my threads and it was all I could take to wait. (I always think you are going to go "This is bull____!")
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  7. #7
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    Quote Originally Posted by tonio View Post
    This is false for n = 1 and k = 3 or 7, or 15 or 31 or....

    Tonio
    Yeah sorry I meant n>1...
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