> Even within number theory, we can easily define an ubounded number of sets

> of finite groups of primes. For example: the number of primes < a hailstone

> number run for a given seed. Or the number of primes less than each perfect

> number as we successively march upward. Or the number of primes less than

> the multiple-valued logs of any given imaginary number. Und so weiter.

May be within number theory, but are still somewhat artificial examples.

It's all just choosing a particular upper bound, and taking all primes

less than it. That's nothing particularly special, no matter where the

upper bound comes from.

But if looking at finite sets, it does no harm to consider all integers,

not just primes. And the class number problem provides a good example

for the original question posed...

Let D<0 be the discriminant of an imaginary quadratic field K, the

simplest kind of algebraic number field other than the rationals. It has

been proved that there are only finitely many D for which the class

number of K is 1, but it is not known exactly what all these

discriminants are. At least not as far as I know. In any case, this is

certainly a highly non-trivial answer to the original question, albeit

with the conditions relaxed slightly. And it remains within number

theory too...

Andy

PS

The values of D which give class number 1 are: -1,-2,-3,-7,-11,-19,

-43,-67 and -163. These values were conjectured by Gauss to be the only

possibilities. It was 1934 before Heilbronn & Linfoot proved that there

were only finitely many such D. Numerical evidence suggests fairly

strongly that the list above is complete. Good old Gauss...