Can you prove that if p_k is prime number, then sum_(k=1)^np_k^-1 is not an integer, for any ninNN?

Can you prove that if ${p}_{k}$ is prime number, then ${\sum}_{k = 1}^{n} {p}_{k}^{-} 1$ is not an integer, for any $n \in \mathbb{N}$?

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Explanation:

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1
Dec 11, 2017

See below.

Explanation:

${\sum}_{k = 1}^{n} \frac{1}{p} _ k = \frac{{\sum}_{k = 1}^{n} \frac{1}{p} _ k {\prod}_{j = 1}^{n} {p}_{j}}{{\prod}_{j = 1}^{n} {p}_{j}} = m$ or

$\left({\sum}_{k = 1}^{n} \frac{1}{p} _ k {\prod}_{j = 1}^{n} {p}_{j}\right) = m {\prod}_{j = 1}^{n} {p}_{j}$

then $\left({\sum}_{k = 1}^{n} \frac{1}{p} _ k {\prod}_{j = 1}^{n} {p}_{j}\right)$ should be divisible by ${p}_{k} , k = 1 , 2 , \cdots , n$ which is an absurd because $\frac{1}{p} _ k {\prod}_{j = 1}^{n} {p}_{j}$ is not divisible by ${p}_{k}$

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