E^pi−pi^e

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It is not known whether 

e π  −  π e

is irrational or rational. ([[e^pi| 

e π

]], known as Gelfond's constant, is a transcendental number. It is not known whether [[pi*e| 

π'e

]], [[pi/e| 

π / e

]], [[2^e| 

2 e

]], [[pi^e| 

π e

]], [[pi^sqrt(2)| 

π  2√  2

]], [[log pi| 

log π

]], Catalan's constant, or the Euler–Mascheroni constant 

γ

are irrational or rational.)

Decimal expansion of e^pi−pi^e

The decimal expansion of 

e π  −  π e

is

$e^{\pi }-\pi ^{e}=0.681534914418223532301934163404812352676791108603519744242043855457416\ldots \,$

giving the sequence of decimal digits (A063504)

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

Continued fraction expansion of e^pi−pi^e

The simple continued fraction expansion of 

e π  −  π e

is

$e^{\pi }-\pi ^{e}=0\,+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{7+{\cfrac {1}{7+{\cfrac {1}{6+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{6+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}\,$

giving the sequence of partial quotients (A063503)

{0, 1, 2, 7, 7, 6, 2, 1, 6, 2, 5, 7, 1, 3, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 11, 5, 6, 2, 1, 124, 1, 4, 2, 1, 1, 3, 18, 1, 1, 1, 1, 17, 1, 2, 10, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2, 4, 84, 1, 1, 1, 4, 1, 1, 15, 2, 1, 1, 17, 1, 1, 8, 1, 1, ...}

See also

• [[pi+e| 

  π  +  e

  ]]
• [[pi−e| 

  π  −  e

  ]]
• [[pi*e| 

  π'e

  ]]
• [[pi/e| 

  π / e

  ]]
• [[pi^e| 

  π e

  ]]
• [[e^pi| 

  e π

  ]] (Gelfond's constant)
• [[e^pi−pi^e| 

  e π  −  π e

  ]]
• [[sqrt(pi*e/2)| 

  2√  π e / 2

  ]]