Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912018553896960 |
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| author | Barnoff, Aaron Bright, Curtis Shallit, Jeffrey |
| author_facet | Barnoff, Aaron Bright, Curtis Shallit, Jeffrey |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_02727 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals Barnoff, Aaron Bright, Curtis Shallit, Jeffrey Formal Languages and Automata Theory Discrete Mathematics Number Theory 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. |
| title | Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals |
| topic | Formal Languages and Automata Theory Discrete Mathematics Number Theory |
| url | https://arxiv.org/abs/2405.02727 |