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Other than the fact that prime numbers are infinite?
jedishrfu said:By rule, you mean one that predicts primes then yes they all fail.
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).
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).
A prime number is a positive integer that is only divisible by 1 and itself. In other words, it has exactly two factors.
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.
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.
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.
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.