Let $\displaystyle p_{i} \ \ \ \ \ 1 \leq i \leq n $ be from the first to the nth prime numbers, prove that $\displaystyle \frac {1}{p_{1}} + \frac {1}{p_{2}} + . . . + \frac {1}{p_{n}}$ is never an integer.

Proof.

I use induction, but I don't know if it is okay.

The first prime number is 2, 1/2 is not an integer, so 1 is okay.

Let $\displaystyle \sum ^{k}_{1} \frac {1}{p_{i}} $ be not an integer.

Then $\displaystyle \sum ^{k+1}_{1} \frac {1}{p_{i}} = \sum ^{k}_{1} \frac{1}{p_{i}} + \frac {1}{p_{k+1}}$ is also not an integer.