The Kempner number is the sum of the series
κ := 2 − 2 0 + 2 − 2 1 + 2 − 2 2 + ⋯ = ∑ n ≥ 0 2 − 2 n . {\displaystyle \kappa :=2^{-2^{0}}+2^{-2^{1}}+2^{-2^{2}}+\cdots =\sum _{n\geq 0}2^{-2^{n}}.}
It is named after Aubrey Kempner, who proved it transcendental in 1916. It is an example of a number easy to prove transcendental which is not a Liouville number.
Properties By definition, the binary expansion of the Kempner number has zeroes everywhere except at places which are powers of two:
κ = 0.110100010000000100000000000000010000000000000000000000000000000100... (base two.) Since the first proof of transcendence by Kempner, many other proofs have been given; see the references. Jeffrey Shallit has proven that it has a simple continued fraction expansion, obtainable by the following construction:
Start with the partial expansion [0, 1, 3]. If the partial expansion is [a, b, ..., y, z], replace it by [a, b, ..., y, z + 1, z − 1, y, ..., b]. If this generated a zero, replace [..., a, 0, b, ...] by [..., a + b, ...]. Repeat steps 2 and 3 indefinitely. This generates the expansion (sequence A007400 in the OEIS)
[ 0 ; 1 , 4 , 2 , 4 , 4 , 6 , 4 , 2 , 4 , 6 , . . . ] = 0 + 1 1 + 1 4 + 1 2 + 1 4 + ⋱ . {\displaystyle [0;1,4,2,4,4,6,4,2,4,6,...]=0+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{2+{\cfrac {1}{4+\ _{\ddots }}}}}}}}}.}
After the first partial quotients, the remainders are all 2, 4 or 6. Since this continued fraction has bounded partial quotients, the Kempner number has irrationality measure 2.
References