Let #P={4,8,9,16,25,cdots,}# be the set of perfect powers, i.e., the set of positive integers of the form #a^b#, where #a,b# are integers greater than #1#. Prove that #sum_(j in P) 1/(j-1) = 1#?

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Nimo N. Share
Feb 24, 2018

Answer:

See below.

Explanation:

"According to Euler, Goldbach showed (in a now lost letter) that the sum of # 1/p − 1 # over the set of perfect powers p, excluding 1 and excluding duplicates, is 1."

"This is sometimes known as the Goldbach-Euler theorem."

Quotes are from article discussing perfect powers at:
https://en.wikipedia.org/wiki/Perfect_power#Examples_and_sums

As for doing the proof, I abstain from attempting it.

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