August 8th, 2010, 05:59 PM  #1 
Senior Member Joined: Nov 2009 Posts: 129 Thanks: 0  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. 
August 8th, 2010, 06:26 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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.

August 8th, 2010, 06:40 PM  #3 
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  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 "oneup" his advice, which is just as good. 
August 8th, 2010, 06:53 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: not an integer Quote:
 
August 8th, 2010, 07:28 PM  #5 
Senior Member Joined: Nov 2009 Posts: 129 Thanks: 0  Re: not an integer
thanks two both 

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