2024/03/29 by Campbell, John M.
#11A67 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2403.20073
The (Mahler-Popken) complexity ‖ n ‖ of a natural number n is the smallest number of ones that can be used via combinations of multiplication and addition to express n, with parentheses arranged in such a way so as to form legal nestings. We generalize ‖ ⋅ ‖ by defining ‖ n ‖m as the smallest number of possibly repeated selections from \ 1, 2, …, m \ (counting repetitions), for fixed m ∈ ℕ, that can be used to express n with the same operational and bracket symbols as before. There is a close relationship, as we explore, between ‖⋅‖2 and lengths of shortest addition chains for a given natural number. This illustrates how remarkable it is that (‖ n ‖2 : n ∈ ℕ ) is not currently included in the On-Line Encyclopedia of Integer Sequences and has, apparently, not been studied previously. This, in turn, motivates our exploration of the complexity function ‖ ⋅‖2, in which we prove explicit upper and lower bounds for ‖⋅‖2 and describe some problems and further areas of research concerning ‖⋅‖2.