Computational Aspects of Sturdy and Flimsy Numbers
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.
Cite
@article{arxiv.2002.02731,
title = {Computational Aspects of Sturdy and Flimsy Numbers},
author = {Trevor Clokie and Thomas F. Lidbetter and Antonio Molina Lovett and Jeffrey Shallit and Leon Witzman},
journal= {arXiv preprint arXiv:2002.02731},
year = {2020}
}