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# Gelfond's constant

Gelfond's constant, named after Aleksandr Gelfond, is

${\displaystyle e^{\pi }={\frac {1}{(i^{i})^{2}}}.\,}$
From Euler's identity,
 e i π + 1 = 0
, we have
${\displaystyle (e^{\pi })^{i}=-1.\,}$

## Decimal expansion of Gelfond's constant

The decimal expansion of Gelfond's constant is

${\displaystyle e^{\pi }=23.1406926327792690057290863679485473802661062426002119934450464095243423506904527835169719970675492\ldots \,}$
A039661 Decimal expansion of
 e π
.
 {2, 3, 1, 4, 0, 6, 9, 2, 6, 3, 2, 7, 7, 9, 2, 6, 9, 0, 0, 5, 7, 2, 9, 0, 8, 6, 3, 6, 7, 9, 4, 8, 5, 4, 7, 3, 8, 0, 2, 6, 6, 1, 0, 6, 2, 4, 2, 6, 0, 0, 2, 1, 1, 9, 9, 3, 4, 4, 5, 0, 4, 6, 4, 0, 9, 5, 2, 4, 3, 4, 2, 3, 5, 0, 6, 9, 0, ...}

## Continued fraction expansion for Gelfond's constant

The simple continued fraction expansion for Gelfond's constant is

${\displaystyle e^{\pi }=23\,+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{591+{\cfrac {1}{2+{\cfrac {1}{9+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}}}}}\,}$
A058287 Continued fraction for
 e π
.
 {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, ...}

## Gelfond's constant - pi

To date, no explanation has been given for why
 e π  −  π
is nearly identical to
 20
.

### Decimal expansion of Gelfond's constant − pi

The fact that

${\displaystyle e^{\pi }-\pi =19.99909997918947576726644298466904449606893684322510617247010181721652\ldots ,\,}$

is almost integer is regarded to be a mathematical coincidence.

A018938 Decimal expansion of
 e π  −  π
.
 {1, 9, 9, 9, 9, 0, 9, 9, 9, 7, 9, 1, 8, 9, 4, 7, 5, 7, 6, 7, 2, 6, 6, 4, 4, 2, 9, 8, 4, 6, 6, 9, 0, 4, 4, 4, 9, 6, 0, 6, 8, 9, 3, 6, 8, 4, 3, 2, 2, 5, 1, 0, 6, 1, 7, 2, 4, 7, 0, 1, 0, 1, 8, 1, 7, 2, 1, 6, 5, 2, 5, 9, 4, 4, 4, 0, ...}

### Continued fraction expansion for Gelfond's constant − pi

The simple continued fraction expansion for
 e π  −  π
is
${\displaystyle e^{\pi }-\pi =19\,+{\cfrac {1}{1+{\cfrac {1}{1110+{\cfrac {1}{11+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{61+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}}}}}\,}$
A018939 Continued fraction for
 e π  −  π
.
 {19, 1, 1110, 11, 1, 2, 2, 2, 2, 1, 61, 3, 2083, 1, 2, 1, 2, 3, 1, 2, 9, 2, 28, 1, 3, 2, 2, 10, 3, 1, 3, 1, 1, 1, 4, 14, 1, 2, 2, 1, 1, 20, 2, 12, 1, 25, 1, 37, 1, 18, 1, 1, 1, 1, 6, 2, 1, 1, 150, 1, 2, 11, 1, 8, 1, 1, 11, ...}

## Square of Gelfond's constant

${\displaystyle e^{2\pi }={\frac {1}{(i^{i})^{4}}}.\,}$

We have

${\displaystyle (e^{2\pi })^{i}=1.\,}$

### Decimal expansion of square of Gelfond's constant

The decimal expansion of the square of Gelfond's constant is

${\displaystyle e^{2\pi }=535.4916555247647365030493295890471814778057976032949155072052550373\ldots \,}$
A216707 Decimal expansion of
 e 2π
.
 {5, 3, 5, 4, 9, 1, 6, 5, 5, 5, 2, 4, 7, 6, 4, 7, 3, 6, 5, 0, 3, 0, 4, 9, 3, 2, 9, 5, 8, 9, 0, 4, 7, 1, 8, 1, 4, 7, 7, 8, 0, 5, 7, 9, 7, 6, 0, 3, 2, 9, 4, 9, 1, 5, 5, 0, 7, 2, 0, 5, 2, 5, 5, 0, 3, 7, 3, 1, 4, 4, 9, 4, 5, 4, 3, 9, 6, ...}