Harmonic Number is not Integer/Proof 1

Proof
As $H_0 = 0$ and $H_1 = 1$, they are integers.

The claim is that $H_n$ is not an integer for all $n \ge 2$.

otherwise:
 * $(\text P): \quad \exists m \in \N: H_m \in \Z$

By the definition of the harmonic numbers:


 * $H_m = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 m$

Let $2^t$ denote the highest power of two in the denominators of the summands.

Then:

Let $S$ be the set of denominators on the of $(2)$.

Then no element of $S$ can have $2$ as a factor, as follows.

Consider an arbitrary summand:


 * $\dfrac {2^{t - 1} } {2^j \times k}$

for some $k \in \Z$, where $j \ge 0$ is the highest power of $2$ that divides the denominator.

For any $2$ to remain after simplification, we would need $j > t - 1$.

Were this to be so, then $2^j \times k$ would have $2^t$ as a factor, and some denominator would be a multiple of $2^t$.

By Greatest Power of Two not Divisor, the set of denominators of the of $(1)$:


 * $\set {1, 2, 3, \ldots, 2^t - 1, 2^t + 1, \ldots, m}$

contains no multiple of $2^t$.

Therefore there can be no multiple of $2$ in the denominators of the of $(2)$.

Let:


 * $\ell = \map \lcm S$

be the lowest common multiple of the elements of $S$.

Because $2$ is not a divisor of any of the elements of $S$, it will not be a divisor of $\ell$.

Hence $\ell$ is likewise odd.

We have:

But the of that last equation is odd, while its  is even.

As this is a contradiction, it follows that our assumption $(\text P)$ that such an $m$ exists is false.

That is, there are no harmonic numbers apart from $0$ and $1$ which are integers.