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Dozens Demigod
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Nov 7 2017, 04:24 AM #217

I suppose the "ugly" sort of semitotative would involve such concoctions as digit-478 base 720, which is 2 * 239. This is of course the most common sort; transparent (e.g. decimal 6) or semitransparent (e.g. dozenal a) totatives are very rare because the neighbours can't ameliorate everything past a certain point. The first time we get a really bad semitotative is hexadecimal e, which is opaque; the first time we get a not only bad but also fairly useless one is probably tetravigesimal m.

The main issue is that there's a difference between single digits and numbers above "10". The former group are simple elements that have to be manipulated, at least if we for simplicity think only about pure bases. The latter group are meant to be broken down additively. It doesn't really matter that 478 is a semicoprime number in decimal because we can work with it simply as 400 + 70 + 8. But when it is a digit, it does matter.

The need to break down numbers to visualise them past a certain point seems to be so innate that I really doubt a base higher than 20 to 30 could ever fly. Even in that range, I presume there would be a lot of sub-base thinking, and the quadratically increasing size of the multiplication table would push them over the line for actual use. The "break" at 30, and the falling out of 24 when we consider regulars as well as divisors, seems to indicate that mathematical considerations are at one with psychological considerations in telling us to go back down the mountain already at about 18 or 20.
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Nov 7 2017, 04:25 AM #218

Why does the "missing last post" bug keep happening? It happened at the start of page 9 and again at the start of page 10. Hopefully this post will fix it.

EDIT: Yes, it did. But can we perhaps look into a more permanent solution that doesn't involve this sort of vacuous post?
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wendy.krieger
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Nov 7 2017, 08:12 AM #219

I'm looking at a rexx script to make the tables, but the sample table (300) seems to have some errors in it. At the moment, i am writing the story-boards, and it looks like this.

Code: Select all

extproc weave.cmd
goto :eof
!topic about

A program to produce the digit-tables as shown on DozenOnline, one for the digits
one for the intuitive division test.

!src
!inc program&#59;  Here is the proggie.
!end

!topic - Numbers

Relative to a base, numbers have a co-composite and a co-prime part.  This is the
regular bit and the period bit,

   coprime classes
      u    coprime = 1 (ie regular)
      a    coprime divides alpha or A2
      w    coprime divides omega or A1
      y    coprime divides A1A2 = R2
      e    other coprimes
   cocomposite class
      n    cocomposite = 1
      r#   cocomposite divides b^#
      s#   cocomposite divides b^{1#},
   general class
      p    prime number
      c    composite number NSFA
      q    number contains a square factor
      x    number is a proper power

No letter is used twice, so it's possible to grep for inclusion or exclusion of a
particular letter.

!topic - Digits of Base $base

The style is to calculate the class-string for each number to the base, and then print
the two tables separately.

                          Digits           Divisibility
      un     unit,        purple           purple
      ur1    divisor      red              red
      ur2    ring 2       pink             shade of red.
      ur3    ring 3        ,,                 ,,
      wn     div A1
      an     div A2
      yn     div R2
      en     coprime
      wr     cp | A1
      ar     cp | A2
      yr     cp | R2
      er     totitive

For the coprimes, there is some sort of shading by grey, as follows.  We either do this
the hard way, or use f1103 to do the hard work.  This would entail constructing a vector
for the base, and then dotting it against successive rows of f1103.  It also means that
we would have to distribute f1103 to users.

     en, i=1    dark grey
     en, i=2    mid grey
     en, i>2    light grey

Since the intent is to evodently find the long periods vide the other ones that the GQR
finds, we could change i=2 into i | 2x

    select
       when i = 1 then dark-grey
       when 2x // i = 0 then mid-grey
       otherwise light-grey

!topic - Meþod of attack.

   Since we have to get the ring of the regular, and the coprime, the easiest way
   is to do successive GCD until 1 is the gcd

   eg     b = 120  n = 7168  gcd = 8  ring = 0
          b =   8  n =  896  gcd = 8  ring = 1
          b =   8  n =  112  gcd = 8  ring = 2
          b =   8  n =   14  gcd = 2  ring = 3
          b =   2  n =    7  gcd = 1  ring = 4

   r4

   Having found this, we then do

    select
       when 7 equals 1      then type = u
       when 7 divides 120-1 then type = w
       when 7 divides 120+1 then type = a
       when 7 divides 120*120-1 then type = y
       otherwise  type = o

    r4w   1/7168 = 0:00 02 01 08 68 68 68 68 68 has 4 leads and a period of 1.

    The co-composite is 7168/7 = 1024,

This happens without having to do any factorisations.

!topic - General Class

This tells us about the number.

   p  Number is prime
   c  Number is composite NSFA
   q  Number contains a square factor
   x  Number is a proper power.



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wendy.krieger
wendy.krieger

Nov 7 2017, 08:20 AM #220

I'm still not sure about how to proceed with the primes.

Over 3/4 of the primes have a period that is derived from Gauss's rule, split between the long, short in slightly equal proportions. Of the non-gaussian primes, we could simply list these at the foot of the table.

In short, it's a matter of deriving the reduced index, which is varying 1, 2, or something bigger. The period of the prime is (p-1)/i. We can accurately uses gauss's rule to find the value is even or odd, and for proper powers, if i | 2x, (where x is the exponent), then the gaussian-fermat rules suffice. For example, for base 16, x = 4, so we reject any value where i divides 8. The first instance is 31, where i=6, and we seek a division into 30 gives a period of 5, rather than the default double-square value of 15.
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Double sharp
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Nov 7 2017, 08:23 AM #221

The different shades of grey for the coprimes indicates the length of their periods, IIRC.

EDIT: Yes, darkest grey is period (p-1), medium grey is period (p-1)/2, and lightest grey is anything shorter (except of course periods 1, 2, and 4, which are coloured as omega, alpha, square-omega, or square-alpha).
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wendy.krieger
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Nov 7 2017, 08:46 AM #222

I was looking at your '300' table, and noticed that neither 91 nor 161 were coloured in some R2 colour. (ie alpha, omega, or the product).

What do you do with composites in general class.

Pouring over icarus's output, i can read most of it. I noticed he had a yates-table in there, that is, primes with periods 1,2,3,4,5... places. I use a different order here, the cunningham style, or true-size. For this process, i will use cunningham, which list the odd numbers just before the double-odd, eg 1,2,4,3,6,8,5,10,12.

At home, i use true-size tables, viz 1,2,6,4,3,10,12,8,5,14,18,9,7.

Code: Select all


           Yates          Carmichael        Size
     1            R      1          R    1         R
     2           11      2         11    2        11
     3          111      4        101    6        R1
     4          101      3        111    4       101
     5        11111      6         R1    3       111
     6           R1      8      10001   10      R0R1
     7      1111111      5      11111   12      RR01
     8        10001     10       R0R1    8     10001
     9      1001001     12       RR01    5     11111
    10         R0R1      7    1111111   14    R0R0R1
    11  11111111111     14     R0R0R1   18    RRR001
    12         RR01     16  100000001    9   1001001

The carmichael system gives an easy to find but well-sorted sizing of the algebraic roots. I've produced an size-sort as far as everything less than 164 digits, but outside of grep, it's rather hard to find the factors. They go quite unimaginably large, but the latest version of factor, seems to handle 120-digit numbers quite well.
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Nov 7 2017, 09:51 AM #223

Eh? 91 and 161 are coloured light blue as alpha-omega mixes. Also, it's Icarus' table, not mine.
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icarus
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Nov 7 2017, 12:12 PM #224

Note: I have had to change my "limits" post, since 2310 is semitotative-dominant, and that is more telling than being highly semitotative.
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Nov 7 2017, 03:23 PM #225

I have filled out base 252. I think for now I will be doing one a day - so Icarus is definitely pulling ahead, which I shall indeed be most grateful for! ^_^
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Nov 7 2017, 04:22 PM #226

I am coding the paragraph. It's just easier & faster that way, bite the bullet. Now it won't take into account grammar for prime powers. The problem with coding language is all the which-switch statements for grammar, oxford commas, etc. and you can get really awkward robot talk. Some of the prose is dry, but I figure it's a good starting point and better than my plucking around for all the pieces. Now I can also get into nitty gritty and count opaque totatives, opaque semitotatives, alpha semitotatives, etc. I figure if you're using the first paragraph as is, I can update it to add the gobbledygook. I also thought of making a little chart of it, but that will require more coding and delay everything. Here's a test of what I have right now. I am going to be madly interrupted all day. Note: I recognize the grammar hiccup, will try to correct so long as it doesn't eat up all the time; moving on to add the other components, but like I said, going to be interrupted as heck today 201711071418: Nearly through but have to drop for the day. Some glitches with a Which statement, but will get through and move ahead this evening or tomorrow. Remaining: list semidivisors, clean up some language, and add the Wieferich, long primes, quadratics. Then things may go more smoothly, since I won't be composing prose each time, but instead have the machine write it directly from the register program. This interpreter is a page long itself; the register is several pages long and the flexCell is longer; all in Wolfram, famous for brief code! 201711072210: Finished it!! Now it should glide on freakin' rails.
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Nov 7 2017, 11:47 PM #227

I'm not actually using the first paragraph as is, but I am certainly using the information from it, as you can see in the latest tours, and spreading it among the "vital statistics" bullet points.
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icarus
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Nov 8 2017, 04:24 AM #228

The cutting and pasting of HTML from the automation means that I can put up a summary in about 5 minutes. The thing that takes the most time is writing the tags for the "Digit Map", etc.! But it is time to turn in. Will hit many more tomorrow. The algorithm isn't perfect, but I don't want to delay things by "fixing" it.
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Nov 8 2017, 04:49 AM #229

icarus @ Nov 8 2017, 04:24 AM wrote: The cutting and pasting of HTML from the automation means that I can put up a summary in about 5 minutes. The thing that takes the most time is writing the tags for the "Digit Map", etc.! But it is time to turn in. Will hit many more tomorrow. The algorithm isn't perfect, but I don't want to delay things by "fixing" it.
Looking forward to it!

I think the automation works best for bases above 120 or maybe 168 because then you can nearly completely forget about the potential civilisational use at any level of civilisation: there simply is none. For bases up to 120 we need to worry about all those tricks like AMT and mixed radices for getting things done.

I guess AMT production can be automated, but I am not sure if its practicality can be. Is the AMT for base 60 practical? Certainly. Is the AMT for base 117 practical? Certainly not. Is the AMT for base 120 practical? It's getting a bit big and Wendy's alternating arithmetic seems better, but I suppose we could handle this simply by declaring that the table shall not have more products than the full multiplication table of hexadecimal or maybe vigesimal. Is the AMT for base 105 practical? It contains a splendid quantity of complementary divisors to leverage all of the first four primes except the one we actually want, 2. Is the AMT for base 21 practical? It gives hardly any leverage, but it is very compact!

For this reason I am focusing on the grand bases first, as they can be done quite a bit more simply. But I haven't forgotten the mid-scale ones: I will soon do up 50, and then I plan to add a bunch of interesting odd bases: {45, 63, 65, 75, 105, 117}. (Feel free to scold me if you think there are some more interesting ones that I am about to unjustly neglect. I know Wendy has mentioned 165 and 195 as having abundant regulars but I don't see why they'd be much different from 105, to be honest, and apart from the detailed approximation for pi, 113 seems totally useless as a large prime.)
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icarus
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Nov 8 2017, 12:40 PM #230

Double sharp: The AMT is automated already. It needs several not-necessarily contiguous small factors less than the square root of the base. How many is not an exact science, but can easily be codified such that we might write code to throw it when necessary. 60, for example, is ideal in that its divisors are neatly cleaved about its square root (as all bases have) with 6 contiguous smallest factors (not all bases have this). That is why AMT is feasible there. AMT is not really applicable to bases the size we are entertaining at the aggregator thread. I do have a prime map like you see in the old school threads, etc. Automated example for base three dozen: I do have some segments written. Divisors, etc. What I had been working on is getting the tables to work directly from the register. This will make them far more flexible than flexCell. Also fully automated is the entire intuitive divisibility test section. (A number x is divisible by 2 if...) This section even incorporates the notion of "trine, nontrine, overtrine, undertrine" (I never want to be undertrine lzozllz), so accommodates nondecimal bases divisible by three. Example of autogenerated intuitive tests, using base 126: (LOTS of impractical there. Even the evenness test might be impractical! The only "practical" tests might be those associated with nonregular numbers!) Here is dozenal: (5 glitched. Easy fix.) What can be more in-depth than has been done here is a description of the semitotative landscape and left/right trim tricks that are "non-intuitive". Semitotatives have been sorted out according to relatedness (coprime factor) and richness (regular factor). Decimal 6, for example, is 2 * 3; it has richness 1 and is omega related. Dozenal ten has richness 1 and alpha-2 related, etc. At some point the tour has to be on its own website. It really sucks a lot of memory here. If it goes there then I will have to credit you for the work you and Oschkar have done. (He is already credited in the base-naming function). As I wrote the tour I have become thoroughly convinced it has to be fully automated. There is just too high of a chance of error if we try to hand-write tables; that is the big problem. Code-generated glitches can be nipped in the bud and corrected for all output but human error or oversight leads to greater, less trackable error. The digit maps are fully automated and that eliminates the necessity to tease out whoopsies. In a similar vein prose can be generated as well. (The summary script I just wrote is sort of a tide-over so that we get forward motion: it can't handle low bases, primes, prime powers, etc., which you are not looking at in the extension).
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Nov 8 2017, 02:59 PM #231

Sure, I think I've finally done and suggested enough to be comfortable with getting credited. It's just that I don't really like my real name to be shown to everyone online, so feel free to PM or email me when the site is closer to going live and I will gladly give it to you (just not post it here). ^_^

I think the impracticality of tests for a base like 126 is a bit of a double question masquerading as a single one. It is absolutely practical if you code it as 7-on-18 or 9-on-14 and absolutely impractical as a pure base. And I am not sure the nonregular tests are as good because there are still 25 multiples of 5 to memorise. I think we can all agree that the test for 125 is practical, but then so is the test for 126; you still look for a trailing zero.

How's the website going to be like? I was thinking that most bases are not actually all that interesting, so we could probably just have the reader type in some number and an automatic chart would come up. But which bases to treat this way?

In the meantime, I think the OP should have 126 and 144 swapped (since 126 is smaller), and I think there's a typo in the title for 168. The newly completed 252 is also missing. Though I wonder if it might not be better to wait until more grand bases are done and fill them all in one fell swoop instead.

P.S. Once the base passes 120, perhaps we could start saying not what auxiliaries the base could use, but what bases could use it as an auxiliary? ^_^ I plan to get to 540 tomorrow and it strikes me that it is a really good angular auxiliary for octodecimal, where it is 1c0{i}. A shame it can only be divided by two twice; perhaps octodecimalists might be more comfortable with the admittedly significantly larger 1080 = 360{i} (lulz!). It really does seem like "360" is always a useful number in the even bases from eight to eighteen.

P.P.S. Looks like this thread has just surpassed a myriad views, which though decimal is pretty cool!
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Nov 18 2017, 04:45 PM #232

Reminder to myself and to the others that the future tours are coming, but I'll have to give a few more of the huge bases a test-drive before I write something. I have 216, 336, and 540 in the works and should be able to post something useful about them shortly.

EDIT: Remaining ones are 150, 160, 192, 216, 288, 300, 336, 432, 540, 600, 630, 660.
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Nov 19 2017, 09:46 PM #233

Code writing has resumed, see this post for the latest. Most of the code work regards tapping the register for data and enveloping it in prose that doesn't sound like robospeak. A tremendous amount of the writing can be automated, and I think (as said many times before) it ought to be, since most of the properties are number-theoretical and easily calculated. The tedium and magnitude of writing an entry, especially tables of dozens or hundreds of colored cells, is a job for the computer and not a poor soul.

It looks like we'll lose HTML capabilities shortly. (Please don't respond to this here, try this thread.) Because of this I am stepping up the web version of the Tour. In a pinch all the code we wrote can be put up at that site.
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Nov 19 2017, 11:42 PM #234

We are also copying over the text that we already have, right? It is pure HTML and there shouldn't be any problem putting those existing entries up with the tables in a pinch. The important thing is to archive all the stuff first. (I mostly put up 50 because I already had it mostly done, but truth be told it's not so interesting; it's like 40 but even clunkier. It just illustrates how the pattern of 18 looks in the heights.)

For a few low bases like 14, 16, 20, 21, and 35, there has been an immense amount of conversation after the main tour posts, and there may be some extra tables past the first page as well that we need to grab. But then again, many of these can be automatically generated. For example, I would like to finally have full multiplication tables for all bases up to 30 and 36 inclusive!! ^_-☆

(The hexadecimal post is an interesting one. It covers the possible civilisational use and would seem to suggest some basic ideas about classifying bases and concerns about civilisationality to be put up too. So I need to gather together a few threads and sort out the material I've written.)
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Kodegadulo
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Mar 1 2018, 10:24 AM #235

Tapatalk has completely trashed icarus's Tour des Bases. All the links in his OP have been junked.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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Kodegadulo
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Mar 2 2018, 01:21 PM #236

Here's Icarus's tour itinerary with the links fixed:
icarus wrote:
But Tapatalk has completely obliterated all the beautiful digit maps and tables icarus developed here. Truly unconscionable.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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icarus
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Jun 28 2018, 10:24 PM #237

Latest developments on the Tour des Bases project.

Focus in June was on graphics; plenty of headway and general efficiency improvements.
baselog12_12_4.png The baselog algorithm can produce radial charts of mantissae of the base-n logarithm of proper regular numbers m | n^e such that n^i does not divide m for ie. This chart pertains to uncial (n = 12) with 1 ≤ e ≤ 4. The mantissae of the base-n logarithms of integers 1 ≤ k < n appear in red and are joined by red irregular polygon. The image is very flexible.
digitdial60.png
The digitDial algorithm produces a radial diagram of the numbers 1 ≤ kn, joining the multiples jd with 1 ≤ jd' such that n = d × d' and coloring the nodes k using the standard color canon. ( = 1, = d | n, = k | n^e with e > 1, = t_α | α with α = n + 1,  = t_ω | ω with ω = n − 1, = 2 with n odd, = t otherwise than stated such that gcd(t, n) = 1, and = k such that k neither divides with e an positive integer nor gcd(k, n) = 1)

There are various fraction illustration diagrams.

Auxiliary base a diagrams such as below:
aux12_144_7.png clock14.png
"Ingredient" pie charts that mesh with the prose that tallies all the species of numbers 1 ≤ kn:
ingredients_120_ght_full.png
New "digit map" routines that are much more flexible than previous ones:
nt_h_30_1.png nt_primes_30_1.png map120_3.png
The left map shows the various semitotatives of bases 2 ≤ n ≤ 30, the center the reptend periods of primes 2 ≤ qn, the right the centovigesimal semitotatives labeled decimally. The center map was generated to n = 2310.

This is a digit map of base n = 510510; it wasn't feasible with the old algorithm and it appears that we can graph bases in the millions. (This number has the maximum number of regular kn for numbers of roughly equivalent magnitude - i.e., it is in A244052, and looks really sparse doesn't it?):
map510510_1.png
We are drawing close to a computer-generated mockup. Once we have a mockup we can tweak the method and then generate the pages en masse. Tasks remaining include prose engines but the graphics engines had seemed to present a greater challenge to code. The pages may be up before a reference glossary. I had written a glossary in October 2012 but the interposing 6 years have more finely honed the mathematical definitions of the various entities we'd defined here between 2008-2011; some of these have been published in cited papers, many entities are now described in the OEIS (i.e., A243822, the "semidivisor counting function", number of numbers k | n^e with 1 ≤ kn and e > 1, the very first, in June 2014).

This June has seen a broad new graphic capability in Mathematica develop that moves away from Cell; the old FlexCell algorithm is very nearly obsolete, but can furnish HTML tables. I am leaning against such tables given the broader visual appeal of the more recent graphics. I am not sure argam will be fully implemented, but may be a second set of pages.

If the project is popular enough and sufficient funds can be raised from page views, I might be able to furnish a dynamic version of part or all of the content, which would be the preferred method, as most of the pictorial content (digit maps, etc.) can already be furnished in a dynamic form. The form would be hosted in a cloud; the funds would cover the cloud fees. But this is a further stage. The dynamic pages would be visually spectacular, and further content could be programmed, i.e., proficiency tests, what-if tools, heuristic content.

Heading into some business these next several weeks so further progress will be limited.
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icarus
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Jul 3 2018, 03:15 AM #238

Here are new color-coded multiplication tables and abbreviated tables.

Sexagesimal AMT:
amt_60_0.png Extended color function.
amt_60_1.png Hexadecimal multiplication table with extended color function:
mt16_1.png Unlabeled, these probably have the same range as sqrt(max(x)) where x is the ability of the numeral map.
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Kodegadulo
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Jul 3 2018, 11:46 AM #239

Major coolness as always, icarus! 😎
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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icarus
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Jul 4 2018, 09:22 PM #240

Kode et al.;

I am hoping to automatically generate a ruler in base b. Now the odd thing about a ruler in any base is that we have to do some ground work assumption about the basis of the b-lengthel. I am okay with dividing the day fully "base"-imally, getting a second-like unit, and then basing the lengthel on the acceleration of earth-gravity. I'd like the lengthel to be close to a foot / 30 cm long. It is not prescriptive, but merely to demonstrate a concept. I don't think the ridiculously large bases would get a lengthel (I hope I am not abusing this word), just the human scale bases.

I like Double sharp's suggestion that bases "in the hundreds" would instead of having auxiliaries, show which bases they could be an auxiliary for, based on the assumptions I'd programmed in the code, i.e., a goal maximum prime, a solid range of small primes brought about by sacrificing large prime factors of bases with gaps among distinct primes.

By the same token, defining a unit of length even for sake of argument would need to describe its fundamentals. The division of time would be presumed to be fully basimal or according to base b auxiliary a. Then we would use acceleration g to arrive at a unit of length ℓ. Maybe we let it lie and not try to cut it to a foot-like length, but if I fail to make it a sensible length, then it could be seen as trying to thumb the scale. Therefore it should be a useful length.
Do you have any suggestions?
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