Harmonic numbers are not integers [Solution]

The problem was to show that

$$H_n = \sum_{k=1}^{n} \frac{1}{k} = 1 + \frac{1}{2} + \dots + \frac{1}{n}$$

is never an integer for $n \gt 1$.

Solution

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There are multiple solutions. One of them uses Bertrand's postulate, which says that there is a prime between $n$ and $2n$ for any $n \gt 1$.

Now consider the largest prime $p$ in $1, 2, \dots, n$.

The denominator of $H_n$ is divisible by $p$. The numerator has terms divisible by $p$, plus the term $1 \times 2 \times \dots \times (p-1) \times (p+1) \times \dots \times n$

By Bertrand's postulate, we have that $2p \gt n$ (otherwise $p$ won't be the largest prime).

Thus the numerator is not divisible by $p$. Thus $H_n$ is not an integer.