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The question is:

Let $n > 1$ be a positive integer. Show that

${1}+{1/2}+1/3 \ldots+1/n$

is not an integer. (Try not to use any powerful results about the distribution of prime numbers)

My attempts:

suppose the $n+1$ th sum is an integer, $k$

$k = 1 + \frac{1}{2} \ldots + \frac{1}{n+1}$

Then the $n$ th sum

$S_{n} = k - \frac{1}{n+1} = \frac{kn+k-1}{n+1}$

The $n$ th sum can also be expressed as

$S_{n} = \frac{a}{n!}$ with some integer $a$

Therefore,

$\frac{a}{n!} = \frac{kn+k-1}{n+1}$

$a = \frac{(n!)(kn+k-1)}{n+1}$

$kn+k-1$ and $n+1$ are coprime so for $a$ to be an integer we need $n+1|n!$

This is where I got stuck. From this I saw that it is then impossible for $n$ to be one less than a prime but from here I don't understand where to go. There are examples of both $n+1$ dividing $n!$ (eg: $n=5$) and of not dividing $n!$.

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    11 different solutions here https://math.stackexchange.com/questions/2746/is-there-an-elementary-proof-that-sum-limits-k-1n-frac1k-is-never-an-int, and many duplicates: https://math.stackexchange.com/questions/linked/2746? – Martin R Feb 20 '24 at 09:17

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