Computer Science > Formal Languages and Automata Theory
[Submitted on 4 May 2024]
Title:Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals
View PDF HTML (experimental)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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