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Computational Fun with Sturdy and Flimsy Numbers

Authors Trevor Clokie, Thomas F. Lidbetter, Antonio J. Molina Lovett , Jeffrey Shallit , Leon Witzman

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ACM Subject Classification
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Dynamic programming
Keywords
  • sturdy number
  • flimsy number
  • context-free grammar
  • finite automaton
  • enumeration

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Abstract

Following Stolarsky, we say that a natural number n is flimsy in base b if some positive multiple of n has smaller digit sum in base b than n does; otherwise it is sturdy . We develop algorithmic methods for the study of sturdy and flimsy numbers.
We provide some criteria for determining whether a number is sturdy. Focusing on the case of base b = 2, we study the computational problem of checking whether a given number is sturdy, giving several algorithms for the problem. We find two additional, previously unknown sturdy primes. We develop a method for determining which numbers with a fixed number of 0’s in binary are flimsy. Finally, we develop a method that allows us to estimate the number of k-flimsy numbers with n bits, and we provide explicit results for k = 3 and k = 5. Our results demonstrate the utility (and fun) of creating algorithms for number theory problems, based on methods of automata theory.

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Trevor Clokie, Thomas F. Lidbetter, Antonio J. Molina Lovett, Jeffrey Shallit, and Leon Witzman. Computational Fun with Sturdy and Flimsy Numbers. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.FUN.2021.10

Author Details

Trevor Clokie
  • University of Waterloo, Canada
Thomas F. Lidbetter
  • University of Waterloo, Canada
Antonio J. Molina Lovett
  • University of Waterloo, Canada
Jeffrey Shallit
  • University of Waterloo, Canada
Leon Witzman
  • University of Waterloo, Canada

Acknowledgements

We would like to thank Kenneth Stolarsky and the referees for helpful comments.

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References

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