Gelfond's constant is simply e raised to the power π. Its transcendence follows from the Gelfond-Schneider theorem (1934): if a is algebraic (≠0, 1) and b is algebraic and irrational, then aᵇ is transcendental. Since e = e¹ and π is transcendental, we can write e^π = (e^(iπ))^(−i) = (−1)^(−i), and since −1 is algebraic and −i is algebraic and irrational, the theorem applies.
This solved Hilbert's 7th problem, one of 23 problems Hilbert posed in 1900 as the most important for 20th-century mathematics. The numerical coincidence e^π − π ≈ 19.999 has no known explanation.