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Math Help - Show that the nth harmonic number is not an integer

  1. #1
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    Show that the nth harmonic number is not an integer



    Let H_n be the nth harmonic number. I started with induction and easily arrived at H_n is not an integer so assume that H_n=\frac{p}{q} where \gcd(p,q)=1. Then H_{n+1}=\frac{p}{q}+\frac{1}{n+1}.

    But I realize now that this is a dead end. How do I proceed? I know I am trying to show that the denominator of the sum is even while the numerator is odd.
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  2. #2
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    This is a common problem.

    Try searching the internet
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  3. #3
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    Re: Show that the nth harmonic number is not an integer

    While the internet is useful, the site you have linked seems to be a collection of rough sketches of proofs; which I had a bit of trouble fleshing out. When I tried to follow similar arguments to construct a proof, I found that things got very messy.

    I did several internet searches before I made my post, but all the arguments I found seemed either: ad-hoc, or dependent on p-adic arguments. And I don't know what p-adic is so...

    I have constructed the following proof. I didn't see anything like this in the online proofs that I found, so I'm worried that my proof is incomplete.

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  4. #4
    Junior Member Sarasij's Avatar
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    Re: Show that the nth harmonic number is not an integer

    If D={1,2}, then m- Ĺ=1=an integer. Itís a trivial case.
    m = r/s + 1/p
    =>(pr+s)/ps=m
    =>s = mps-pr
    =>s=p(ms-r)
    i.e. p|s which is a contradiction due to the defn of s. This might suffice the proof. Your approach was excellent.
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