Gelfond's constant

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In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e raised to the power π. Like both e and π, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that

where i is the imaginary unit. Since −i is algebraic but not rational, eπ is transcendental. The constant was mentioned in Hilbert's seventh problem.[1] A related constant is , known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.[2]

Numerical value[edit]

The decimal expansion of Gelfond's constant begins


If one defines and

for , then the sequence[3]

converges rapidly to .

Geometric property[edit]

The volume of the n-dimensional ball (or n-ball), is given by

where is its radius, and is the gamma function. Any even-dimensional ball has volume

and, summing up all the unit-ball (R = 1) volumes of even-dimension gives[4]

See also[edit]


  1. ^ Tijdeman, Robert (1976). "On the Gel'fond–Baker method and its applications". In Felix E. Browder. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.1. American Mathematical Society. pp. 241–268. ISBN 0-8218-1428-1. Zbl 0341.10026. 
  2. ^ Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I. 322 (10): 909–914. Zbl 0859.11047. 
  3. ^ Borwein, J.; Bailey, D. (2004). Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters. p. 137. ISBN 1-56881-211-6. Zbl 1083.00001. 
  4. ^ Connolly, Francis. University of Notre Dame[full citation needed]

Further reading[edit]

External links[edit]