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Yes, seems like there are only finitely many words over a binary alphabet that do not contain a non-primitive word (0, 01, 010 and 1, 10, 101). How would it cha
by Gehinnn 1y ago
Yes, seems like there are only finitely many words over a binary alphabet that do not contain a non-primitive word (0, 01, 010 and 1, 10, 101).
How would it change if the alphabet has three symbols?
- n4r9 1y agoIt certainly seems like you can get much longer words. I just had a quick go and came up with 0 1 0 2 0 1 2 0 1 0 2 0 2 1 but I stopped there because it gets tedious to check manually for repetition. Might be worth writing a little script to produce the word where each letter is the smallest possible number that doesn't create repetition.
- Gehinnn 1y agoJust checked with AI: Thue showed 1906 that there are infinitely many square free words (:= a word that doesn't contain a non-primitive word) over an alphabet with at least 3 symbols.
- n4r9 1y agoCool! This paper is also quite readable: https://arxiv.org/pdf/2104.04841 https://arxiv.org/pdf/2104.04841 On p.2 they follow my idea of adding the lowest possible letter at the end, although they generalise it to adding the letter as close to the end as possible. They conjecture that this process does not stop. I'm always amazed with combinatorics how quickly you arrive at questions that no one knows the answer to.