Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 LiouvilleNumber

Source Notebook

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:

In[1]:=
ResourceFunction["LiouvilleNumber"][]
Out[1]=

Compute the Liouville number to 403 digits:

In[2]:=
ResourceFunction["LiouvilleNumber"][403]
Out[2]=

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

In[3]:=
ResourceFunction["LiouvilleNumber"][14/33, 1096]
Out[3]=

Properties and Relations (2) 

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

In[4]:=
ContinuedFraction[ResourceFunction["LiouvilleNumber"][200]]
Out[4]=

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

In[5]:=
ResourceFunction["LiouvilleNumber"][25] - Root[21 - 190 # - 75 #^3 + 10 #^6 &, 1]
Out[5]=

Neat Examples (1) 

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

In[6]:=
DiscretePlot[ ResourceFunction["LiouvilleNumber"][1/b, MachinePrecision], {b, 2, 16}, PlotRange -> All]
Out[6]=

Publisher

Peter Burbery

Version History

  • 1.0.0 – 30 August 2022

Related Resources

License Information