Is it true that any rule regarding prime numbers eventually fails?

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Discussion Overview

The discussion revolves around the nature of rules or formulas related to prime numbers, specifically whether any such rule eventually fails in predicting or generating primes. The scope includes theoretical considerations and mathematical reasoning regarding the properties of prime numbers.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that any rule predicting primes ultimately fails, emphasizing that this applies to regularities comprehensible to humans and not to probabilistic statements like the Prime Number Theorem.
  • Others challenge this view, arguing that there are formulas that can generate all prime numbers, questioning the validity of the claim that all rules fail.
  • A participant seeks clarification on whether a formula can take a prime number as input and return a binary result indicating primality, referencing Wilson's Theorem as a potential example.

Areas of Agreement / Disagreement

Participants express differing opinions on the validity of rules regarding prime numbers, with some claiming all such rules fail while others argue that certain formulas successfully generate primes. The discussion remains unresolved with multiple competing views.

Contextual Notes

Some claims depend on specific definitions of "rule" and "regularity," and there is ambiguity regarding the types of formulas being discussed, particularly in relation to their inputs and outputs.

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Other than the fact that prime numbers are infinite?
 
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By rule, you mean one that predicts primes then yes they all fail.
 
jedishrfu said:
By rule, you mean one that predicts primes then yes they all fail.

Not just prediction of prime numbers, but any regularity that humans can comprehend. The regularity must be directly operated on primes rather than on another set of numbers, such as the statement that every even number is the sum of two primes, or every even number is the sum of a prime and a semi prime. Also, the regularity can't be probabilistic as in the Prime Number Theorem.
 
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No, it is not true. How could it be?

This page has many examples of formulas that generate all the primes (and only the primes).
 
eigenperson said:
No, it is not true. How could it be?

This page has many examples of formulas that generate all the primes (and only the primes).

Do any of them take prime numbers as inputs? :confused:
 
I don't quite understand your question -- are you asking for something like a formula that takes a number as input, and returns 1 if it is prime and 0 if it is non-prime?

If so, take a look at Wilson's Theorem (or at the first formula on the page I just linked to).
 
eigenperson said:
I don't quite understand your question -- are you asking for something like a formula that takes a number as input, and returns 1 if it is prime and 0 if it is non-prime?

If so, take a look at Wilson's Theorem (or at the first formula on the page I just linked to).

Wilson's formula is very interesting. :biggrin: Thanks for the info.
 
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