Is there possibly a largest prime number? Prime numbers are numbers with no factors other than one and itself.
Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have.
So the larger the number, it seems like the less likely the number is to be a prime.
Surely there must be a number where, simply, every number above it has some other factors.  A "critical point" where every number larger than it simply will always have some factors other than one and itself.
Has there been any research as to finding this critical point, or has it been proven not to exist?  That for any $n$ there is always guaranteed to be a number higher than $n$ that has no factors other than one and itself?

Now that this question has been bumped up, I feel like posting the other somewhat famous proof that they are infinitely many primes: Consider the Fermat numbers $F_n = 2^{2^n} + 1$. It is an easy exercise to see that the gcd of any two Fermat numbers is $1$. As each number has at least one prime factor, picking for each $F_n$ a factor $p_n$ gives an infinite sequence of prime numbers.
This proof is usually attributed to Pólya, but it may well be much older. See this discussion of its history on Math Overflow.
