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August 8th, 2010, 06:59 PM   #1
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not an integer

let n be integers in Z with n>1. Show that ?(^n)(_i=1) (1/i) is not an integer. So i am trying to prove that 1/i is not an integer.
I put some number in to see if this is correct and it is. however i don't know how to generalize it if n >1 for all n.

thanks.
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August 8th, 2010, 07:26 PM   #2
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Re: not an integer

You're trying to show that H_1 = 1 is the only integer harmonic number. The key step is looking at the denominator and seeing how many primes divide it.
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August 8th, 2010, 07:40 PM   #3
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Re: not an integer

Hi again tinynerdi.

It's pretty clear that for all n > 1, 1/n is not an integer, right? Because no positive integer greater than 1 divides 1.

Now, for the sum, you probably want to use induction on n. Clearly, if n = 2, then



is not an integer. Now, assume that for some n > 2, the statement is true, i.e. suppose that



is not an integer. Clearly, it is a rational number, though, so let's say



in lowest terms. We want to show that is not an integer. But of course



Now, see if you can prove that the above fraction cannot be an integer. [Hint: Assume the contrary and derive a contradiction.]

Best of luck.


Addendum. Sorry, I noticed just after I posted that CRGreathouse had already replied. I did not mean to try to "one-up" his advice, which is just as good.
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August 8th, 2010, 07:53 PM   #4
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Re: not an integer

Quote:
Originally Posted by Ormkärr
Addendum. Sorry, I noticed just after I posted that CRGreathouse had already replied. I did not mean to try to "one-up" his advice, which is just as good.
Yours is much more useful; I was in too much of a hurry to give a real answer (but figured I'd at least leave a hint),
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August 8th, 2010, 08:28 PM   #5
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Re: not an integer

thanks two both
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