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Gelfond–Schneider constant
The Gelfond–Schneider constant, named after Aleksandr Gelfond and Theodor Schneider, is
-
2 √ 2.
The Gelfond–Schneider constant was proved to be a transcendental number by Rodion Kuzmin in 1930.
Contents
Decimal expansion of the Gelfond–Schneider constant
The decimal expansion of the Gelfond–Schneider constant is
-
2 √ 2≈ 2.6651441426902251886502972498731398482742113137146594928359795933649204461787059548676091800051964169419893638542353...
giving the sequence of decimal digits (see A007507)
- {2, 6, 6, 5, 1, 4, 4, 1, 4, 2, 6, 9, 0, 2, 2, 5, 1, 8, 8, 6, 5, 0, 2, 9, 7, 2, 4, 9, 8, 7, 3, 1, 3, 9, 8, 4, 8, 2, 7, 4, 2, 1, 1, 3, 1, 3, 7, 1, 4, 6, 5, 9, 4, 9, 2, 8, 3, 5, 9, 7, 9, 5, 9, 3, 3, 6, 4, 9, 2, 0, 4, 4, 6, 1, 7, 8, 7, 0, 5, 9, 5, 4, ...}
Binary expansion of the Gelfond–Schneider constant
The binary expansion of the Gelfond–Schneider constant is
-
2 √ 2≈ 10.1010101001000110111000101111001111111011000000000110001011100011000101101100011000101110110111100100000111001000001...2
giving the sequence of binary digits (see A??????)
- {1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, ...}
Continued fraction expansion of the Gelfond–Schneider constant
The continued fraction expansion of the Gelfond–Schneider constant is
2 √ 2 = 2 +
|
giving the sequence of integer part and partial denominators (see A062979)
- {2, 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 2, 9, 1, 2, 1, 4, 1, 1, 6, 4, 8, 1, 6, 2, 1, 1, 1, 1, 1, 5, 1, 6, 1, 1, 2, 2, 6, 68, 1, 3, 3, 4, 10, 8, 4, 1, 16, 10, 1, 1, 3, 1, 25, 2, 3, 2, 1, 3, 6, 2, ...}
Square root of the Gelfond–Schneider constant
The square root of the Gelfond–Schneider constant gives
- √ 2= (2√ 2√ 2)
= (21 2
)1 2 √ 2=√ 2√ 2,
- √ 2= 2√ 2
= 2√ 22
=1 √ 2.√ 2√ 2
The square root of the Gelfond–Schneider constant is a transcendental number.
This shows that an irrational number (transcendental number) to the power of an irrational number (irrational algebraic number) can sometimes produce a rational number, since this number raised to the power of√ 2 |
-
( √ 2√ 2)√ 2= (√ 2) 2 = 2.
Decimal expansion of the square root of the Gelfond–Schneider constant
The decimal expansion of the square root of the Gelfond–Schneider constant is
- √ 2=√ 2√ 2√ 2=≈ 1.6325269194381528447734953810247196020791088570531141172477806843830352059986166422478555075066260414...√ 2√ 2
giving the sequence of decimal digits (see A078333)
- {1, 6, 3, 2, 5, 2, 6, 9, 1, 9, 4, 3, 8, 1, 5, 2, 8, 4, 4, 7, 7, 3, 4, 9, 5, 3, 8, 1, 0, 2, 4, 7, 1, 9, 6, 0, 2, 0, 7, 9, 1, 0, 8, 8, 5, 7, 0, 5, 3, 1, 1, 4, 1, 1, 7, 2, 4, 7, 7, 8, 0, 6, 8, 4, 3, 8, 3, 0, 3, 5, 2, 0, 5, 9, 9, 8, 6, 1, 6, 6, 4, 2, ...}
Binary expansion of the square root of the Gelfond–Schneider constant
The binary expansion of the square root of the Gelfond–Schneider constant is
- √ 2=√ 2√ 2√ 2=≈ 1.1010000111101101010010001100000011010011100101010000111001010001001111101001000110000111101010010101...2√ 2√ 2
giving the sequence of binary digits (see A??????)
- {1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, ...}
n n √ n
The Gelfond–Schneider constant may be viewed as the following special case:
-
n n √ n, n = 2.
n-th root of n n √ n
- n√ n= (nn√ nn√ n) 1 / n = (n 1 / n )n√ n= (n√ n)n√ n,
n, n ≥ 2, |
This shows that an irrational number (transcendental number) to the power of an irrational number (irrational algebraic number) can sometimes produce a rational number, since this number raised to the power of
( n √ n ) n − 1 |
n |
-
( n√ n)n√ n(n√ n) n − 1 = (n√ n) (n√ n) n = (n√ n) n = n.
Also,
-
( n√ n) (n√ n) k (n√ n) n − k = (n√ n) (n√ n) n = (n√ n) n = n, 1 ≤ k ≤ n − 1.
bn, k ^ en, k = n |
bn, k |
en, k |
* When
n |
bn, k |
** When
n |
en, k ∈ {( 4√ 4 ) 2, (16√ 16 ) 8, (64√ 64 ) 32, ...} ∪ {(27√ 27 ) 9, (729√ 729 ) 243, (19683√ 19683 ) 6561, ...} ∪ {(256√ 256 ) 64, ...} ∪ {(3125√ 3125 ) 625, ...} ∪ ⋯ |
For example,
- 3√ 33√ 3(3√ 3) 2 = (3√ 3) (3√ 3) 3 = (3√ 3) 3 = 3.
Also,
- 3√ 3(3√ 3) 23√ 3= (3√ 3) (3√ 3) 3 = (3√ 3) 3 = 3.
See also
External links
- Weisstein, Eric W., Gelfond–Schneider Constant, from MathWorld—A Wolfram Web Resource.
- Weisstein, Eric W., Gelfond’s Theorem, from MathWorld—A Wolfram Web Resource.