This paper clears up to the following three conjectures: 1.The conjecture of Ehle [1] on the A-acceptability of Padé approximations to e^z, which is true; 2.The conjecture of Nørsett [5] on the zeros of the E-polynomial, which is false; 3.The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist. We further give necessary as well as sufficient conditions for A-stable (acceptable) rational approximations, bounds for the highest order of restricted Padé approximations and prove the non-existence of A-acceptable restricted Padé approximations of order greater than 6. The method of proof, just looking at order stars and counting their fingers, is very natural and geometric and never uses very complicated formulas.