Prove by induction that $1+\frac{1}{2}+\frac{1}{3} +...+\frac{1}{n}$ for any $1<n$ and $n\in \Bbb{N}$ is not a natural number.
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Without induction: $\sum_{i=1}^n \frac{1}{i}=k$, then $\sum_{i=1}^n \frac{n!}{i}=kn!$. By Bertrand's Postulate exists a prime $n/2<p<n$. But $p$ divides everything except $n!/p$, contradiction. – user236182 Oct 04 '15 at 17:05
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1@user236182 "It is a very strange phenomenon that many problem books seem to push the Bertrand's Postulate solution to this problem. I remember that this came up as a problem (apropos of nothing) in my freshman year math class, and I had some problem book at hand and duly turned in a solution which used BP. The next year I got the problem in a number theory course and by then was sophisticated enough to see the elementary solution involving the ord_2 function. – Pete L. Clark Aug 19 '10 at 0:04" – Aloizio Macedo Oct 04 '15 at 17:06