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

In summary, the conversation discusses the concept of regularities that can be applied to prime numbers. It is stated that any regularity that humans can comprehend must directly operate on primes and cannot be probabilistic. It is also mentioned that there are formulas that can generate all and only prime numbers, such as Wilson's Theorem.
  • #1
goldust
89
1
Other than the fact that prime numbers are infinite?
 
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  • #2
By rule, you mean one that predicts primes then yes they all fail.
 
  • #3
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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  • #4
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).
 
  • #5
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:
 
  • #6
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).
 
  • #7
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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1. What is a prime number?

A prime number is a positive integer that is only divisible by 1 and itself. In other words, it has exactly two factors.

2. What is the rule regarding prime numbers that eventually fails?

The rule that eventually fails is the belief that there exists a finite list of prime numbers that can be used to determine if any given number is prime or not. This rule is known as the Prime Number Theorem.

3. Why does this rule eventually fail?

This rule eventually fails because there is no known pattern or formula that can accurately predict the distribution of prime numbers. Prime numbers are essentially random and infinite, making it impossible to create a finite list or formula to determine all prime numbers.

4. Are there any other rules or methods for determining prime numbers?

Yes, there are several other methods for determining prime numbers, such as the Sieve of Eratosthenes, the AKS primality test, and the Lucas-Lehmer test. However, these methods are also limited and cannot accurately determine all prime numbers.

5. Does the failure of this rule have any implications in mathematics?

Yes, the failure of this rule has significant implications in mathematics, particularly in the field of number theory. It challenges our understanding of the nature of prime numbers and highlights the complexity and mystery surrounding them.

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