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Math Help - A Relation About Bernoulli Numbers and Theorem of Arithemtic

  1. #1
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    A Relation About Bernoulli Numbers and Theorem of Arithemtic

    Hi,

    As I posted in to number theory forum, I noticed that

    <br />
N^{s+1}\equiv N \quad Denominator[B_s]<br />

    where B_s is the s-th Bernoulli Number

    I think that i have also a correct proof.

    Here I also have the opportunity to thank this forum not olny for the help prividing me, but also for giving me the chance of expressing my concerns in mathematics.
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    Last edited by gdmath; October 23rd 2009 at 05:57 AM. Reason: Upload Revisited document
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by gdmath View Post
    Hi,

    As I posted in to number theory forum, I noticed that

    <br />
N^{s+1}\equiv N \quad Denominator[B_s]<br />

    where B_s is the s-th Bernoulli Number

    I think that i have also a correct proof.

    Here I also have the opportunity to thank this forum not olny for the help prividing me, but also for giving me the chance of expressing my concerns in mathematics.
    I haven't read all the way through this yet, but I have read through your proof of your test for series convergence.

    In the test of convergence your statement of the test and notation needs to be improved.

    It could read something like:

    Test of Convergence

    The series \sum_{i=1}^{\infty} a_n converges if:

    \lim_{n\to \infty} \frac{a_n a_{n-1}}{a_n-a_{n-1}}=0

    Also I cannot follow your argument here:

    A Relation About Bernoulli Numbers and Theorem of Arithemtic-gash.png

    Please inprove the notation and if possible clarify the reasoning.

    (I will leave you post here for two weeks for you to prot a revision after that I will removing it as the standard of presentation is at present not acceptable)

    CB
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  3. #3
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    About the test of convergence i based on ratio test proof as i found it here :
    http://calculus7.com/sitebuildercont...fratiotest.rtf

    At the point of

    <br />
\sum_{n=1}^{\infty}\frac{f(i)}{p-n\cdot p^{n-1}\cdot f(i)}<br />

    I mean that i have a new series with close form:

    <br />
a_n=\frac{f(i)}{p-n\cdot p^{n-1}\cdot f(i)}<br />

    and i prove that this series is convergent throught ratio test.

    Since a_n>f(i+n) , then throught the comparison test if a_n converges then also f(i+n) converges.


    However my doubt is at the beggining, when i solve only the right inequality:

    <br />
-r<\frac{f(j)\cdot f(j+1)}{f(j+1)-f(j)}<r<br />

    Please tell me if there is a fault point (because i do not concern about the left inequality).


    Anyway the convergence test might not belong in this paper (beyond the impovments that proof needs).

    I include it in the corrected paper only if we are 100% certain that proof is correct and proper typed.


    Thank you a lot.
    Last edited by gdmath; October 21st 2009 at 01:09 AM.
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  4. #4
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    revisited

    I have made a revision of the document.

    Should i wait this thread to be deleted (and post new one) or leave it here?
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    Last edited by gdmath; August 26th 2009 at 12:31 AM.
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