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?

Another proof is:
Consider the numbers $$9^{2^n} + 1, \\ \\  n = 1,2,\dots$$
Now if $$9^{2^n} + 1 = 0 \mod p$$ then we have that, for $ m > n$ that
$$9^{2^m} + 1 = (9^{2^n})^{2^{m-n}} + 1 = (-1)^{2^{m-n}} + 1 = 1+1 = 2 \mod p$$
Thus if one term of the sequence is divisible by a prime, none of the next terms are divisible by that prime, i.e. if you write out the factors of the terms of the sequence, each term of this sequence gives rise to a prime not seen before!
As a curiosity, it can be shown that each number in the sequence has at least one prime factor > 40. See this question on this very site: Does $9^{2^n} + 1$ always have a prime factor larger than $40$?
