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arXiv · 2405.02727

Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals

Aaron Barnoff·Curtis Bright·Jeffrey Shallit
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Abstract

We show that the $n$'th digit of the base-$b$ representation of the golden ratio is a finite-state function of the Zeckendorf representation of $b^n$, and hence can be computed by a finite automaton. Similar results can be proven for any quadratic irrational. We use a satisfiability (SAT) solver to prove, in some cases, that the automata we construct are minimal.

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Aaron Barnoff, Curtis Bright, Jeffrey Shallit. 2024-05-04. Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals. https://doi.org/10.1007/978-3-031-71112-1_3

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