Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is $eπ$, that is, $e$ raised to the power $\pi$. 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

$$ e^\pi = (e^{i\pi})^{-i} = (-1)^{-i},$$

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

Numerical value
The decimal expansion of Gelfond's constant begins


 * $$e^\pi = $$ $23.141$...

Construction
If one defines $k0 = 1⁄√2$ and

$$k_{n+1} = \frac{1 - \sqrt{1 - k_n^2}}{1 + \sqrt{1 - k_n^2}}$$

for $n > 0$, then the sequence

$$(4/k_{n+1})^{2^{-n}}$$

converges rapidly to $eπ$.

Continued fraction expansion
$$e^{\pi} = 23+ \cfrac{1} {7+\cfrac{1} {9 +\cfrac{1} {3+\cfrac{1} {1+\cfrac{1} {1 +\cfrac{1} {591+\cfrac{1} {2+\cfrac{1} {9+\cfrac{1} {1+\cfrac{1} {2+\cfrac{1} {\ddots} }           }            }            }                  }               }            }         }      }   } $$

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

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

As given by the integer sequence A058287.

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

$$V_n = \frac{\pi^\frac{n}{2}R^n}{\Gamma\left(\frac{n}{2} + 1\right)},$$

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

$$V_{2n} = \frac{\pi^n}{n!}R^{2n},$$

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

$$\sum_{n=0}^\infty V_{2n} (R = 1) = e^\pi.$$

Ramanujan's constant
$$e^{\pi{\sqrt{163}}} = (\text{Gelfond's constant})^{\sqrt{163}}$$

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

Similar to $eπ - π$, $eπ√163$ is very close to an integer:


 * $$e^{\pi \sqrt{163}} = $$ $262,537,412,640,768,740$... $$ \approx 640\,320^3+744$$

This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fool article in Scientific American magazine, "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 $6403203 + 744$ is explained by complex multiplication and the q-expansion of the j-invariant, specifically:

$$j((1+\sqrt{-163})/2)=(-640\,320)^3$$

and,

$$(-640\,320)^3=-e^{\pi \sqrt{163}}+744+O\left(e^{-\pi \sqrt{163}}\right)$$

where $O(e-π√163)$ is the error term,

$${\displaystyle O\left(e^{-\pi {\sqrt {163}}}\right) = -196\,884/e^{\pi {\sqrt {163}}}\approx -196\,884/(640\,320^{3}+744)\approx -0.000\,000\,000\,000\,75}$$

which explains why $eπ√163$ is 0.000 000 000 000 75 below $6403203 + 744$.

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

The number $eπ − π$
The decimal expansion of $eπ − π$ is given by A018938:


 * $$e^{\pi} - \pi = $$ $19.999$...

This is approximately equal to:

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

That said, as viewed from the framework of algorithmic information theory, it is not hard to form a large set of short arithmetic expressions using well known constants and available operators, where some expression values turn out to land very near to an integer. With enough fluidity in what is considered the allowable expression set, this kind of thing can begin to resemble data dredging.

The number $πe$
The decimal expansion of $πe$ is given by A059850:



\pi^{e} = $$ $19.999$...

It is not known whether or not this number is transcendental. Note that, by Gelfond-Schneider theorem, we can only infer definitively that $ab$ is transcendental if $22.459$ is algebraic and $a$ is not rational ($b$ and $a$ are both considered complex numbers, also $a ≠ 0$, $a ≠ 1$).

In the case of $eπ$, we are only able to prove this number transcendental due to properties of complex exponential forms, where π is considered the modulus of the complex number $eπ$, and the above equivalency given to transform it into $(-1)-i$, allowing the application of Gelfond-Schneider theorem.

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

The number $eπ − πe$
As with $πe$, it is not known whether $eπ − πe$ is transcendental. Further, no proof exists to show whether or not it is irrational.

The decimal expansion for $eπ − πe$ is given by A063504:



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

The number $ii$
Using the principal value of the complex logarithm, $$i^{i} = (e^{i\pi/2})^i = e^{-\pi/2} = (e^{\pi})^{-1/2}$$

The decimal expansion of is given by A049006:



i^{i} = $$ $0.682$...

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:

$0.208$ is both algebraic (a solution to the polynomial $x2 + 1 = 0$), and not rational, hence $ii$ is transcendental.