Function Repository Resource:

# LiouvilleNumber

Compute the Liouville number or its generalization

Contributed by: Peter Cullen Burbery
 ResourceFunction["LiouvilleNumber"][] computes the Liouville number to machine precision. ResourceFunction["LiouvilleNumber"][p] computes the Liouville number to precision p. ResourceFunction["LiouvilleNumber"][r,p] computes the Liouville number corresponding to the rational number r to precision p.

## Details

The number r must be between 0 and 1.
The generalized Liouville number is a transcendental number defined as for a rational number r=p/q.
The regular Liouville number corresponds to r=1/10.

## Examples

### Basic Examples (2)

Compute the Liouville number:

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Compute the Liouville number to 403 digits:

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Compute the Liouville number corresponding to 14/33 to 1096 digits:

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### Properties and Relations (2)

Large integers sporadically appear in the continued fraction expansion of the Liouville number:

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The Liouville number is very nearly equal to the root of a certain sixth-degree polynomial:

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### Neat Examples (1)

Plot values of the first few Liouville numbers corresponding to 1/b:

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Peter Burbery

## Version History

• 1.0.0 – 30 August 2022