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

OEISA039661

Construction[edit]

If one defines and

for , then the sequence[3]

converges rapidly to .

Demonstration:[edit]

The sequence {}

=

Continued fraction expansion[edit]

This is based on the digits for the simple continued fraction:

As given by the integer sequence A058287.

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]

Gelfond's constant - [edit]

The decimal expansion of is given by A018938:

Despite this being nearly the integer 20, no explanation has been given for this fact and it is believed to be a mathematical coincidence.

Similar or related constants[edit]

Ramanujan's constant[edit]

This is known as Ramanujan's constant. It is an application of Heegner numbers, where 163 is the Heegner number in question.

Similar to , is very close to an integer:

As it was the Indian mathematician Srinivasa Ramanujan who first predicted this almost-integer number, it has been named after him, though the number was first discovered by the French mathematician Charles Hermite in 1859.

The coincidental closeness, to within 0.000 000 000 000 75 of the number is explained by complex multiplication and the q-expansion of the j-invariant, specifically:

and,

where is the error term,

which explains why is 0.000 000 000 000 75 below .

(For more detail on this proof, consult the article on Heegner numbers.)

The number [edit]

The decimal expansion of is given by A059850:

It is not known whether or not this number is transcendental. Note that, by Geldond-Schneider theorem, we can only infer definitively that is transcendental if is algebraic and is not rational (and are both considered complex numbers).

In the case of , we are only able to prove this number transcendental due to properties of complex exponential forms, where is considered the modulus of the complex number , and the above equivalency given to transform it into , allowing the application of Gelfond-Schneider theorem.

has no such equivalence, and hence, as both and are transcendental, we can make no conclusion about the transcendence of .

The number [edit]

As with , it is not known whether is transcendental. Further, no proof exists to show whether or not it is irrational.

The decimal expansion for is given by A063504:

The number [edit]

The decimal expansion of is given by A049006:

Because of the equivalence, we can use Gelfond-Schneider theorem to prove that the reciprocal square root of Gelfond's constant is also transcendental:

is both algebraic (a solution to the polynomial ), and not rational, hence is transcendental.

See also[edit]

References[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]