Gelfond's constant explained

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is, that is, raised to the power . Like both and, 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

$$e^\backslash pi\; =\; (e^)^\; =\; (-1)^,$$

where is the imaginary unit. Since is algebraic but not rational, 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  +  is also irrational.^[2]

Numerical value

The decimal expansion of Gelfond's constant begins

e ^{\pi} = ...  

Construction

If one defines and

$$k\_\; =\; \backslash frac$$

for, then the sequence^[3]

$$(4/k\_)^$$

converges rapidly to .

Continued fraction expansion

$$e^\; =\; 23+\backslash cfrac$$

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

$$e^\; =\; [23;\; 7,\; 9,\; 3,\; 1,\; 1,\; 591,\; 2,\; 9,\; 1,\; 2,\; 34,\; 1,\; 16,\; 1,\; 30,\; 1,\; 1,\; 4,\; 1,\; 2,\; 108,\; 2,\; 2,\; 1,\; 3,\; 1,\; 7,\; 1,\; 2,\; 2,\; 2,\; 1,\; 2,\; 3,\; 2,\; 166,\; 1,\; 2,\; 1,\; 4,\; 8,\; 10,\; 1,\; 1,\; 7,\; 1,\; 2,\; 3,\; 566,\; 1,\; 2,\; 3,\; 3,\; 1,\; 20,\; 1,\; 2,\; 19,\; 1,\; 3,\; 2,\; 1,\; 2,\; 13,\; 2,\; 2,\; 11,\; ...]$$

As given by the integer sequence A058287.

Geometric property

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

$$V\_n\; =\; \backslash frac,$$

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

$$V\_\; =\; \backslash fracR^,$$

and, summing up all the unit-ball volumes of even-dimension gives^[4]

$$\backslash sum\_^\backslash infty\; V\_\; (R\; =\; 1)\; =\; e^\backslash pi.$$

Similar or related constants

Ramanujan's constant

$$e^\; =\; (\backslash text)^$$

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:

e ^{\pi} } = ... ≈ 640320 ^{3+744}

This number was discovered in 1859 by the mathematician Charles Hermite.^[5] In a 1975 April Fool article in Scientific American magazine,^[6] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.

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:

$$j((1+\backslash sqrt)/2)=(-640\backslash ,320)^3$$

and,

$$(-640\backslash ,320)^3=-e^+744+O\backslash left(e^\backslash right)$$

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

The decimal expansion of is given by A018938:

e ^{\pi} - \pi = ...

This is approximately equal to:

The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows: $$\backslash sum\_^\backslash left(8\backslash pi\; k^2\; -2\; \backslash right)\; e^\; =\; 1.$$ The first term dominates since the sum of the terms for

k\geq 2 total \sim 0 . 0003436 . The sum can therefore be truncated to \left( 8 \pi -2 \right) e ^{-\pi} ≈ 1 , where solving for e ^{\pi} gives e ^{\pi} ≈ 8 \pi -2 . Rewriting the approximation for e ^{\pi} and using the approximation for 7 \pi ≈ 22 gives $$e^\; \backslash approx\; \backslash pi\; +\; 7\backslash pi\; -\; 2\; \backslash approx\; \backslash pi\; +\; 22-2\; =\; \backslash pi+20.$$Thus, rearranging terms gives e ^{\pi} - \pi ≈ 20 . Ironically, the crude approximation for 7 \pi yields an additional order of magnitude of precision.^[7]

The number

The decimal expansion of is given by A059850:

\pi ^{e} = ...

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

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

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:

e ^{\pi} - \pi ^{e} = ...

The number

Using the principal value of the complex logarithm,$$i^\; =\; (e^)^i\; =\; e^\; =\; (e^)^$$

The decimal expansion of is given by A049006:

i ^{i} = ...

Because of the equivalence, we can use the 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

• Transcendental number
• Transcendental number theory, the study of questions related to transcendental numbers
• Euler's identity
• Gelfond–Schneider constant

Further reading

• Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press, 2007,

External links

• Gelfond's constant at MathWorld
• A new complex power tower identity for Gelfond's constant
• Almost Integer at MathWorld

Notes and References

1. Book: Tijdeman, Robert . Felix E. Browder . Felix Browder . Mathematical Developments Arising from Hilbert Problems . . XXVIII.1 . 1976 . . 0-8218-1428-1 . Robert Tijdeman . On the Gel'fond–Baker method and its applications . 241–268 . 0341.10026 .
2. Nesterenko, Y. Yuri Valentinovich Nesterenko. Modular Functions and Transcendence Problems. Comptes Rendus de l'Académie des Sciences, Série I. 322. 909–914. 1996. 10. 0859.11047 .
3. Book: Jonathan Borwein . Borwein . J. . Bailey . D. . Mathematics by Experiment: Plausible Reasoning in the 21st Century. limited . Wellesley, MA. A K Peters. 2004. 137 . 1-56881-211-6 . 1083.00001 .
4. Connolly, Francis. University of Notre Dame
5. Book: Barrow , John D . The Constants of Nature . Jonathan Cape . 72 . 2002 . London . 0-224-06135-6 .
6. Gardner . Martin . Mathematical Games . Scientific American . 232 . 4 . 127 . April 1975 . Scientific American, Inc . 10.1038/scientificamerican0575-102 . 1975SciAm.232e.102G .
7. [Eric Weisstein]