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. When n is proven flimsy by multiplier k, we say n is k-flimsy. We study computational aspects of sturdy and flimsy numbers.We provide some criteria for determining whether a number is sturdy. We study the computational problem of checking whether a given number is sturdy, giving several algorithms for the problem, focusing particularly on the case b=2.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.