Counting function for powers of primes

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Discussion Overview

The discussion revolves around a proposed method for counting functions related to powers of primes, touching on its validity and potential implications in number theory. The scope includes theoretical exploration and practical testing of the method through programming.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that the method should be tested with a program to verify its effectiveness, noting that the problem is an open one in number theory.
  • Another participant claims that the proposed method is generally correct and can yield accurate results, but expresses skepticism about the possibility of significantly condensing the expressions involved.
  • It is mentioned that a formula for the nth prime would be necessary for further simplification, and that the same approach could be applied to the prime counting function, which may not have a simple formula.
  • A participant expresses regret over the removal of a previous post, indicating that it contained interesting content.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the feasibility and implications of the proposed method, and the discussion remains unresolved.

Contextual Notes

Limitations include the need for a formula for the nth prime and the potential complexity of the prime counting function, which may not be easily simplified.

robnybod
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at this point it is more important for you to show that your method works. I suggest you write a program and test your method using the formulas you posted. If the test works for large numbers, then you have something serious. By the way, this problem is an open problem in number theory so you may just have come up with a solution.
 
"Method" is correct and works, in the sense that it will give correct numbers in general. Unfortunately you will not be able to condense your expressions significantly. You would then, among other things, likely need a formula for the nth prime, and btw the exact same set up of inclusion/exclusion and floors can be used to express the prime counting function, for which there is very likely no short and elementary formula.
 
Sorry to see you remove this. Was Interesting.
 

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