The greatest prime number

If q is the greatest prime number, then there are only finitely many primes, certainly no more than q-1. Therefore they can be numbered: p(1), p(2), ..., p(m), with m less than q-1, of course. The properties of the number

L:= p(1) p(2)... p(m) q + 1

are quite paradoxical. Being greater than q, it is not prime. On the other hand, none of the numbers p(1), p(2), ..., p(m), q being its divisor, L has no natural divisors other than 1 and itself, so it is prime. Therefore there must be some error in its definition. The only uncertain moment in the definition, however, was the assumption that there is a greatest prime number. Hence it is this assumption that must be false.

Historically speaking, this is the first mathematical theorem which claims that there is infinitely many of something. It was proved about 2300 years ago, most probably by Euclid.

Notwithstanding, more or less scientific journals bring now and then new information that the greatest prime number is - to give a real example of January, 1994 - the number 2^{859433}-1. Obviously, what is meant here is that this is the greatest prime number that we can write down at the given moment e.g. in the decimal system. Searching for such numbers seems to be a good job. Indeed, there is no end to it.