Transcendence Theory - Open Problems

Open Problems

While the Gelfond-Schneider theorem proved that a large class of numbers was transcendental, this class was still countable. Many well known mathematical constants are still not known to be transcendental, and in some cases it is not even known whether they are rational or irrational. A partial list can be found here.

A major problem in transcendence theory is showing that a particular set of numbers is algebraically independent rather than just showing that individual elements are transcendental. So while we know that e and π are transcendental that doesn't imply that e + π is transcendental, nor other combinations of the two (except eπ, Gelfond's constant, which is known to be transcendental). Another major problem is dealing with numbers that are not related to the exponential function. The main results in transcendence theory tend to revolve around e and the logarithm function, which means that wholly new methods tend to be required to deal with numbers that cannot be expressed in terms of these two objects in an elementary fashion.

Schanuel's conjecture would solve the first of these problems somewhat as it deals with algebraic independence and would indeed confirm that e+π is transcendental. It still revolves around the exponential function however and so would not necessarily deal with numbers such as Apéry's constant or the Euler–Mascheroni constant. Another extremely difficult unsolved problem is the so-called Constant or Identity problem.