This article is available for purchase or by subscription. See below.
Abstract
|
Define
to be the
complexity
of
, the smallest number
of 1s needed to write
using an arbitrary combination of addition and multiplication. John Selfridge showed
that
for
all
,
leading this author and Zelinsky to define the
defect of
,
, to be the
difference
.
Meanwhile, in the study of addition chains, it is common to consider
, the number of small
steps of
, defined
as
, an integer
quantity, where
is the length of the shortest addition chain for
. So here we
analogously define
,
the
integer defect of
,
an integer version of
analogous to
.
Note that
is not
the same as
.
We show that
has additional meaning in terms of the defect well-ordering we considered in 2015, in that
indicates which
powers of
the quantity
lies between when
one restricts to
with
lying in a specified congruence
class modulo
. We also
determine all numbers
with
,
and use this to generalize a result of Rawsthorne (1989).
|
PDF Access Denied
We have not been able to recognize your IP address
44.212.94.18
as that of a subscriber to this journal.
Online access to the content of recent issues is by
subscription, or purchase of single articles.
Please contact your institution's librarian suggesting a subscription, for example by using our
journal-recommendation form.
Or, visit our
subscription page
for instructions on purchasing a subscription.
You may also contact us at
contact@msp.org
or by using our
contact form.
Or, you may purchase this single article for
USD 40.00:
Keywords
integer complexity, well-ordering, arithmetic formulas
|
Mathematical Subject Classification 2010
Primary: 11A67
|
Milestones
Received: 2 August 2018
Revised: 7 April 2019
Accepted: 29 April 2019
Published: 23 July 2019
|
|