It is well known that the Harmonic numbers
$$H_n=\sum_{k=1}^n \frac{1}{k}$$
are never integers for $n>1$.
Can the difference of two Harmonic numbers
$$H_n-H_m=\sum_{k=m+1}^n \frac{1}{k}$$
be an integer if $n>m>1$?
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1If I am not wrong it can not be integer, because you can take the highest power of $2$ that the denominators contain and notice that there is only one fraction which denominator contains that power of $2$. – pointer Dec 31 '14 at 13:38
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You should also be able to prove it by induction on the number of terms in the sum. – G. H. Faust Dec 31 '14 at 13:56