An interesting and very much open question in number theory. The real numbers are a vector space (of infinite dimension, even of uncountable dimension!) over the field of rational numbers. As such, we can ask whether any set of real numbers is linearly independent. *Is the set {1,e,π} linearly independent?*

That is, do there exist integers a,b such that a*e+b*π is an integer?

We don't know.

Can we get it with a=b? That is, *"is e+π a rational number? Is e-π?"*

We don't know.

Of course, it would be absurd if the answer to any of these questions was "yes". I doubt any mathematician seriously countenances the possibility. But we have nothing remotely like a proof.

What about algebraic dependence? *Is there a polynomial with integer coefficients P(x,y)∈***Z**[x,y] for which P(e,π)=0?

We don't know.

It is fairly easy to prove irrationality of e and of various powers of e. Niven's proof of the irrationality of pi is completely elementary. A theorem of Lindemayer tells us that various powers of e are even transcendental numbers. ζ(3)=∑_{n≥1}1/n^{3} is known to be irrational. Since e and π are transcendental, so are e+sqrt(2), e-17sqrt(3), π*sqrt(5), and the like; in particular, we know they're irrational.

And that's about all we know. Proving irrationality is hard.