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Henderson · #11 January 27th, 2008, 09:42 AM

Let $A_{n} = 1+\frac{1}{2}+...+\frac{1}{n}$ .

Assume that, for some $n$ , $A_{n}$ is an integer.

Let $P$ be the product of all the denominators except for the largest prime less than $n$ . $P$ is clearly an integer.

Then $P \cdot A_{n} = P \cdot (1+\frac{1}{2}+...+\frac{1}{n})$ .

On the LHS, $P \cdot A_{n}$ is an integer, based on how we defined P and our assumption.

On the RHS, after distributing, each term becomes an integer except for the term with the largest prime as the denominator. This clearly is not an integer, so the RHS is the sum of many integers plus one non-integer, and the RHS as a whole is not an integer.

Left with the integer LHS equaling a non-integer RHS, our assumption must be false, and $A_{n}$ must not be an integer.

$Q.E.D.$

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