The greatest prime number
If q is the greatest prime number, then there are only finitely
many primes, certainly no more than q1. Therefore they can be numbered:
p(1), p(2), ..., p(m), with m less than q1, 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.
M.K.
