I'll admit up front that this post makes me definitely guilty of a near-repetition of something I posted earlier. This time, I think I have a clearer phrasing of what I want -- hence the repetition.
Does anyone know of any real numbers r which have the following two properties?
1) r is known to be algebraic. 2) No one knows an explicit polynomial over Z for which r is a root.
Hmmm.. I thought that was a really clear statement of what I'm trying to find. But it isn't because someone could say "r where r = the truth value (either 0 or 1) of the statement of the Goldbach conjecture."
But that's not the type of thing I mean. I'm thinking of numbers which are given by an explicit series like eta(3).