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?

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≥11/n3 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.