Gelfond's constant is e raised to the power π. Its approximate value is 23.14069263277927… Proving it transcendental was Hilbert's 7th problem, posed in 1900 as one of the 23 most important unsolved questions for the 20th century. Alexander Gelfond solved it in 1934.
e^π sits tantalizingly close to 23 but misses by 0.14. The coincidence e^π - π ≈ 19.999 is even closer but equally meaningless.
The Gelfond-Schneider theorem (1934) states: if a is algebraic, not 0 or 1, and b is algebraic and irrational, then a^b is transcendental. Gelfond's constant e^π = (e^(iπ))^(−i) = (−1)^(−i). Here a = −1 (algebraic) and b = −i (algebraic and irrational). The theorem applies directly.
Table showing examples of numbers proved transcendental by Gelfond-Schneider
| Expression | a | b | Result |
|---|---|---|---|
| e^π = (-1)^(-i) | -1 | -i | transcendental |
| 2^√2 (Hilbert) | 2 | √2 | transcendental |
| √2^√2 | √2 | √2 | transcendental |
The numerical near-miss e^π − π ≈ 19.9990999 has no known mathematical explanation. It is likely a coincidence, but similar coincidences (like Ramanujan's constant) sometimes turn out to have deep reasons. e^π has been computed to millions of decimal places: 23.14069263277926900572908636794854738…
e^π > π^e. This can be proved without a calculator: the function x^(1/x) has a maximum at x=e, so e^(1/e) > π^(1/π), which gives e^π > π^e.
Gelfond's constant e^pi ≈ 23.14069. Proving it transcendental was Hilbert's 7th problem (1900). Gelfond solved it in 1934: if a is algebraic (not 0 or 1) and b is algebraic and irrational, then a^b is transcendental. Since e^pi = (-1)^(-i), and -1 and -i are algebraic with -i irrational, the theorem applies. The near-coincidence e^pi - pi ≈ 19.999 has no known mathematical explanation.