[FOM] Prime values of polynomials

[joeshipman@aol.com]

I have read that no integer-coefficient polynomials of degree >1 are 
known to take infinitely many prime values; conjecturally, all the 
irreducible ones do.

This is a nice example, but it's not so easy to tell whether a 
polynomial is irreducible.

Can anyone provide a comparably simple example of a property which is 
believed to hold for all integers, but which is not known to hold for 
any? (Alternatively, I'll accept an example of a property which is 
conjectured to hold for all members of a set X but is not known to hold 
for any, where X is easier to recognize than "irreducible polynomials 
of degree >1" even if X is not as easy to recognize as the integers 
are).

-- JS