Question

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)^np_k^-1` is not an integer, for any `ninNN`?

Answer 1

See below.

Explanation:

`sum_(k=1)^n 1/p_k = (sum_(k=1)^n 1/p_k prod_(j=1)^n p_j)/(prod_(j=1)^n p_j) = m` or

`(sum_(k=1)^n 1/p_k prod_(j=1)^n p_j) = m prod_(j=1)^n p_j`

then `(sum_(k=1)^n 1/p_k prod_(j=1)^n p_j)` should be divisible by `p_k, k = 1,2,cdots,n` which is an absurd because `1/p_k prod_(j=1)^n p_j` is not divisible by `p_k`