×

Hello! Socratic's Terms of Service and Privacy Policy have been updated, which will be automatically effective on October 6, 2018. Please contact hello@socratic.com with any questions.

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#?

1 Answer
Dec 11, 2017

Answer:

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#