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E^pi−pi^e

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It is not known whether 
eπ  −  πe
is irrational or rational. ([[e^pi| 
eπ
]], known as Gelfond's constant, is a transcendental number. It is not known whether [[pi*e| 
π'e
]], [[pi/e| 
π / e
]], [[2^e| 
2e
]], [[pi^e| 
πe
]], [[pi^sqrt(2)| 
π
2  2
]], [[log pi| 
log π
]], Catalan's constant, or the Euler–Mascheroni constant
γ
are irrational or rational.[1][2][3])

Decimal expansion of e^pi−pi^e

The decimal expansion of 
eπ  −  πe
is

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 
eπ  −  πe
is

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| 
    πe
    ]]
  • [[e^pi| 
    eπ
    ]] (Gelfond's constant)
  • [[e^pi−pi^e| 
    eπ  −  πe
    ]]
  • [[sqrt(pi*e/2)| 
    2  πe / 2 
    ]]

Notes

  1. Weisstein, Eric W., Pi, from MathWorld—A Wolfram Web Resource.
  2. Weisstein, Eric W., Irrational Number, from MathWorld—A Wolfram Web Resource.
  3. Some unsolved problems in number theory