Challenges of creating custom numeration systems and automatic sequences
Linfoot Number Theory Seminar
10th May 2023, 11:00 am – 12:00 pm
Fry Building,
A correct choice of a system of representing the integers can often simplify the analysis of a particular problem. In this talk, first, we will set up the necessary vocabulary, see what is required to define a numeration system. With a handful of interesting questions from additive number theory in mind, we check how some of them are successfully solved by choosing an appropriate numeration system, and some that might require defining a custom numeration system and what are the challenges of defining it.
We mainly work with Automatic sequences which are the sequences that look like 011010011001...; that have fractal-like self-similarity. In general, we check if certain interesting properties and conjectures hold for these sequences. Whilst we use a new (technically, a decade old) software called ‘Walnut’ designed to “automatically” prove results about automatic sequences, we are required to choose a numeration system in which Walnut can interpret the input integers; we stick to natural numbers for the purposes of this talk. Where obvious, we readily use bases like binary or decimal. Otherwise, we can define a specific numeration system like Fibonacci base which we will see as a running example throughout and discuss some caveats of making such a numeration system work (computational or otherwise).
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