Iterates of increasing sequences of positive integers |
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Authors: | Vichian Laohakosol Boonrod Yuttanan |
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Institution: | 1. Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok, 10900, Thailand 2. Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Songkhla, 90112, Thailand
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Abstract: | It is proved that for fixed integers ${K \ge 2, D \ge 2, {\rm if} (D - 1) \mid K}$ , then there exists a unique increasing sequence (a(n)) n ≥ K of positive integers such that $$\underset{K {\rm times}}{\underbrace{a ( a ( \dots a(a}}(n)) \dots)) = Dn$$ otherwise, there are uncountably many increasing sequences of positive integers (a(n)) satisfying this iterated functional equation. This generalizes recent results of Propp and Allouche–Rampersad–Shallit. |
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