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arXiv · 2002.02731

Computational Aspects of Sturdy and Flimsy Numbers

Trevor Clokie·Thomas F. Lidbetter·Antonio Molina Lovett·Jeffrey Shallit·Leon Witzman
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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 Molina Lovett, Jeffrey Shallit, Leon Witzman. 2020-02-07. Computational Aspects of Sturdy and Flimsy Numbers. https://arxiv.org/abs/2002.02731

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