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Nathan Addy29-Jan-07 12:37
Nathan Addy29-Jan-07 12:37 
It can; you just can't check it directly (that is, by checking each number).

For instance, even though there are infinitely many natural numbers, you can prove that there are an infinite number of primes, but only a finite number of even primes:

Proof there are an Infinite number of primes > 1:
Assume there are a finite number of primes, and call them p_1, p_2, ..., p_n.
Think of the number p_1 * p_2 * ... * p_n + 1. This number is bigger than any of the primes, and so it can't be a prime itself. But if you try to divide that number by any of the primes, you'll always end up with a remainder of one, and so the new number must be a prime itself. But that is a contradiction, because we assumed the only primes were p_1 through p_n. Because our hypothesis leads to a contradiction, it must be false. Thus, there must be infinitely many primes.

Proof there are a finite number of even primes:
Any even number is divisible by 2. Because 2 != any even number greater than 2, any even number greater than 2 is divisible by a number that is neither equal to 1 nor to itself. Since there are only finitely many even numbers equal to 2 or less (only one, which is two itself), there are only a finite number of even primes.

Basically, by using different techniques that let you talk about abstract sets of numbers, including infinite sets, you can prove things about infinite sets of things, without having to check each and every number individually (which is of course impossible).

So even though it hasn't been proven mathematically (yet) whether or not there are infinitely many twin primes, it should be possible. (Although there is a famous mathematical theorem that says that there are mathematically true statements that cannot be proven, so there is a slim chance that this theorem might be one of those. On the other hand, nothing like this has been found yet, outside of some very abstract statements that were used to prove that very theorem itself, so most likely a proof can be found.)
QuestionScoring a number of values/weightings and normalizing Pin
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QuestionSuper Fast edit-distance algorithm: Sift2 [modified] Pin
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QuestionNeed help with "!" sign in postfix parser Pin
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