Pi^e

This article needs more work.

Please help by expanding it!

π^e (pi^e) is conjectured to be transcendental, while e^π (e^pi), known as Gelfond's constant, is a transcendental number. (It is not known whether pi*e, pi/e, 2^e, pi^e, pi^sqrt(2), log pi, Catalan's constant, or the Euler–Mascheroni constant $\scriptstyle \gamma \,$ are irrational.)

Decimal expansion of pi^e

The decimal expansion of

π  e

is

π  e  =  22.4591577183610454734271522045437350275893151339966...

giving the sequence of decimal digits (A059850)

{2, 2, 4, 5, 9, 1, 5, 7, 7, 1, 8, 3, 6, 1, 0, 4, 5, 4, 7, 3, 4, 2, 7, 1, 5, 2, 2, 0, 4, 5, 4, 3, 7, 3, 5, 0, 2, 7, 5, 8, 9, 3, 1, 5, 1, 3, 3, 9, 9, 6, 6, 9, 2, 2, 4, 9, 2, 0, 3, 0, 0, 2, 5, 5, 4, 0, 6, 6, 9, 2, 6, 0, 4, 0, 3, 9, 9, 1, 1, ...}

Continued fraction expansion for pi^e

The simple continued fraction expansion of

π  e

is

$\pi ^{e}=22\,+{\cfrac {1}{2+{\cfrac {1}{5+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}}}}}\,$

or, in a more compact form

$\pi ^{e}=[22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,4,1,2,1,1,50,1,1,1,1,7,1,1,1,3,6,1,20,10,1,2,10,1,8,2,2,1,1,1,4,1,43,2,2,3,1,\ldots ],\,$

giving the sequence (A058288)

{22, 2, 5, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 9, 15, 25, 1, 1, 5, 4, 1, 2, 1, 1, 50, 1, 1, 1, 1, 7, 1, 1, 1, 3, 6, 1, 20, 10, 1, 2, 10, 1, 8, 2, 2, 1, 1, 1, 4, 1, 43, 2, 2, 3, 1, 2, 8, 1, 1, 16, 1, 4, 1, 3, 1, 1, 1, 2, 1, 1, 6, 1, 2, ...}

See also

• π+e (pi+e)
• π−e (pi−e)
• π*e (pi*e)
• π/e (pi/e)
• π^e (pi^e)
• e^π (e^pi) (Gelfond's constant)
• e^π−π^e (e^pi−pi^e)