Numbers with Integer Complexity Close to the Lower Bound
Abstract
Define
|n|
to be the complexity of
n
, the smallest number of 1's needed to write
n
using an arbitrary combination of addition and multiplication. John Selfridge showed that
|n|≥3
log
3
n
for all
n
. Define the defect of
n
, denoted
δ(n)
, to be
|n|−3
log
3
n
; in this paper we present a method for classifying all
n
with
δ(n)<r
for a given
r
. From this, we derive several consequences. We prove that
|
2
m
3
k
|=2m+3k
for
m≤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
δ(n)<r
and
n≤x
, we prove that
A
r
(x)=
Θ
r
((logx
)
⌊r⌋+1
)
, allowing us to conclude that the values of
|n|−3
log
3
n
can be arbitrarily large.