Using finite automata to compute the base-b representation of the golden ratio and other quadratic irrationals
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 bn, 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.
0
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.