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Gelfond's constant
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Gelfond's constant, named after Aleksandr Gelfond, is
e i π + 1 = 0 
Gelfond's constant is a transcendental number.
Contents
Decimal expansion of Gelfond's constant
The decimal expansion of Gelfond's constant is
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
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 whye π − π 
20 
Decimal expansion of Gelfond's constant − pi
The fact that
is almost integer is regarded to be a mathematical coincidence.
A018938 Decimal expansion ofe 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 fore π − π 
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
The square of Gelfond's constant is
We have
Decimal expansion of square of Gelfond's constant
The decimal expansion of the square of Gelfond's constant is
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, ...}
See also

Ramanujan's constant = e π √ 163= (e π )√ 163= (Gelfond's constant)√ 163 
π e 
e π − π e 
π + e 
π − e 
π ⋅ e 
π /e