9853

Nested Structure of the Continued Fraction of Liouville's Constant

The Liouville constant is the transcendental number defined by . Computing its continued fraction expansion in this semilog plot of the number of digits reveals the presence of sporadic large terms and a nested structure.

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The notation means an integer with 9's.
Interestingly, the incrementally largest term (excluding the term ) occurs precisely at positions , and this term consists of ( 9's.
To compute the entries of this expansion the use of high-precision arithmetic is required; in this calculation, 500000-digit precision was used.
The more common way to represent a continued fraction is as follows:
but this is too wide to display for many terms.
This structure was found by the author.
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