Suppose I told you that the Goldbach conjecture is one of those. (The Goldbach conjecture supposes that every even integer number can be expressed as the sum of two odd primes.)

Is that logically possible? (And, no, I haven't proved it!)

It seems to me that, as Steven poined out earlier, there are an infinite number of even integers.  If we are to prove the GC to be true, we must test each and every instance, which would be impossible.  Someone could, theoretically, DISPROVE the statement by finding an exception... but no one has.