Numbers with Integer Complexity Close to the Lower Bound

Abstract

Define knk to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that knk 3 log 3 n for all n. Define the defect of n, denoted by ı.n/, to be knk 3 log 3 n; in this paper we present a method for classifying all n with ı.n/ < r for a given r. From this, we derive several consequences. We prove that k2 m 3 k k D 2m C 3k for m Ä 21 with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m. Furthermore, defining A r .x/ to be the number of n with ı.n/ < r and n Ä x, we prove that A r .x/ D ‚ r ..log x/ brcC1 /, allowing us to conclude that the values of knk 3 log 3 n can be arbitrarily large.

Integer complexity: Representing numbers of bounded defect

“…One can then write down theorems about A r and B r similar to those above and in [4] and [1] about A r and B r . We will state them here without proof, as the proofs are the same except for the strictnesses of some of the inequalities.…”

“…In this section we review the results of [4] and [1] regarding the defect δ(n), the stable complexity n st and stable defect δ st (n) described below, and low-defect polynomials.…”

“…(4) n ∈ T α , where T α is as defined in [4] (and thus in particular either n = 1 or n = n − 1 + 1.) (5) There is a good factorization n = u · v with u ∈ T α and v ∈ B α .…”

“…For instance, one outstanding question regarding integer complexity, raised by Guy [9], is that of the complexity of 3-smooth numbers; is 2 n 3 k always equal to 2n + 3k, whenever n and k are not both zero? This author and Zelinsky used the study of the defect to show in [4] that this holds true whenever n ≤ 21.…”

“…It is not monotonic in s; for instance, let us consider what happens as s approaches 1 from below. We use the classification of numbers of defect less than 1 from [4]. For any s < 1, there's a finite set of numbers m such that any n with δ(n) < 1 can be written as m = n3 k for some k. Or in other words, T s necessarily consists of a finite set of constants.…”