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Number Theory - Show harmonic numbers are not integers

  1. Nov 29, 2005 #1
    Q.Prove that 1+1/2+1/3+1/4+.........+1/n is not an integer.n>0
     
  2. jcsd
  3. Nov 29, 2005 #2
    There are a couple ways to prove this that I know of. Your title says number theory so if you have discussed in class (or read on your own) p-adic valuations, try to show that for a fixed n>1 the 2-adic valuation is always greater than 1. If you don't have experience with this, try the sum where n=p (some prime). Make an argument (manipulate terms) why this isnt' an integer. Then fix that p, and pick an x between p and 2p-1, and follow a similar argument. Unfortunately you now must prove that for any x such a p exists. I believe this was proven by Erdos. Maybe your teacher will let you get away with that as justification. Another way is to show that for some n, the sum=(even number + odd number)/2N *odd number which isn't an integer.
     
  4. Nov 29, 2005 #3

    AKG

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    I think you mean n > 1. If it's just n > 0, then the sum for n = 1 should not be an integer, but that sum is just 1, which clearly is an integer.
     
  5. Dec 2, 2005 #4

    CarlB

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    Is there a theorem that you know that says that (for large enough m), there is always a prime number between m and 2m?

    Carl
     
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