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Pages: (10) « First ... 8 9 [10] ( Go to first unread post ) 
Double sharp 
Posted: Nov 7 2017, 04:24 AM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
I suppose the "ugly" sort of semitotative would involve such concoctions as digit478 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 subbase 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. 
Double sharp 
Posted: Nov 7 2017, 04:25 AM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
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? 
wendy.krieger 
Posted: Nov 7 2017, 08:12 AM


Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
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 storyboards, and it looks like this.


wendy.krieger 
Posted: Nov 7 2017, 08:20 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
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 nongaussian 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 (p1)/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 gaussianfermat 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 doublesquare value of 15. 
Double sharp 
Posted: Nov 7 2017, 08:23 AM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
The different shades of grey for the coprimes indicates the length of their periods, IIRC.
EDIT: Yes, darkest grey is period (p1), medium grey is period (p1)/2, and lightest grey is anything shorter (except of course periods 1, 2, and 4, which are coloured as omega, alpha, squareomega, or squarealpha). 
wendy.krieger 
Posted: Nov 7 2017, 08:46 AM


Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
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 yatestable in there, that is, primes with periods 1,2,3,4,5... places. I use a different order here, the cunningham style, or truesize. For this process, i will use cunningham, which list the odd numbers just before the doubleodd, eg 1,2,4,3,6,8,5,10,12. At home, i use truesize tables, viz 1,2,6,4,3,10,12,8,5,14,18,9,7.
The carmichael system gives an easy to find but wellsorted sizing of the algebraic roots. I've produced an sizesort 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 120digit numbers quite well. 

Double sharp 
Posted: Nov 7 2017, 09:51 AM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
Eh? 91 and 161 are coloured light blue as alphaomega mixes. Also, it's Icarus' table, not mine.

icarus 
Posted: Nov 7 2017, 12:12 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Note: I have had to change my "limits" post, since 2310 is semitotativedominant, and that is more telling than being highly semitotative.

Double sharp 
Posted: Nov 7 2017, 03:23 PM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
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!

icarus 
Posted: Nov 7 2017, 04:22 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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 whichswitch 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. Base 432  Quadringentoduotrigesimal (Dozenal: Base 300;  Trinabiqual) The number 432 is neither squarefree nor a prime power with 2 distinct prime divisors {2, 3}. Its prime decomposition is 2^{4} × 3^{3} and has the prime signature “43”, the smallest base to have such signature. Four hundred thirty two number. Four hundred thirty two has 20 divisors {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, 432}, 20 regular numbers in all. Base 432 has 144 totatives (1 out of 3 digits, 33.3%), and there are 81 prime totatives. The quadringentoduotrigesimal ωnumber is 431, a prime.. The αnumber is 433, also prime. Both ω and α prime makes for a “bone dry“ set of quadringentoduotrigesimal intuitive divisibility tests that cover 2 of the 8 smallest primes. There are 269 neutral digits, of which 12 are semidivisors (richness 2: 32, 64, 81, 96, 128, 162, 192, 243, 256, 288, 324, 384) and 257 are semitotatives (10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, …). Of the 257 semitotatives, there are 28 alpha2related semitotatives and 229 opaque semitotatives. Long primes in base 432 include 5, 7, 17, and 19. In base 432, 5 is a thirdorder Wieferich prime. There are 44 quadringentoduotrigesimal quadratic residues. Base 210  Ducentodecimal (Dozenal: Base 156;  Unpenthexal) The number 210 is squarefree with 4 distinct prime divisors {2, 3, 5, 7}. Its prime decomposition is 2 × 3 × 5 × 7 and has the prime signature “1111”, the smallest base to have such signature. Two hundred ten is a primorial that also is the seventeenth highly regular and the thirteenth highly semidivisible number. Two hundred ten has 16 divisors {1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210}, 16 regular numbers in all. Base 210 has 48 totatives (8 out of 35 digits, 22.9%), and there are 42 prime totatives. The ducentodecimal ωnumber is 209, which is divisible by 11 and 19. The αnumber is 211, a prime. The base has omegadominant intuitive divisibility tests that cover the 5 smallest and further, 6 of the smallest 8 primes. There are 147 neutral digits, of which 52 are semidivisors (quantities by richness starting with 2: 25, 14, 7, 3, 2, 1) and 95 are semitotatives (22, 26, 33, 34, …). Of the 95 semitotatives, there are 23 omegarelated semitotatives and 72 opaque semitotatives. Long primes in base 210 include 13 and 17. There are 48 ducentodecimal quadratic residues. Base 120  Centovigesimal (Dozenal: Base a0;  Decanunqual) The number 120 is neither squarefree nor a prime power with 3 distinct prime divisors {2, 3, 5}. Its prime decomposition is 2^{3} × 3 × 5 and has the prime signature “311”, the smallest base to have such signature. One hundred twenty is a that also is the fifth superior highly composite, tenth a highly composite, and the fourteenth highly regular number. One hundred twenty has 16 divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120}, 16 regular numbers in all. Base 120 has 32 totatives (4 out of 15 digits, 26.7%), and there are 27 prime totatives. The centovigesimal ωnumber is 119, which is divisible by 7 and 17. The αnumber is 121, the square of 11. Base 120 thus has a relatively rich set of intuitive divisibility tests that cover the 5 smallest and further, 6 of the smallest 8 primes. There are 73 neutral digits, of which 20 are semidivisors (richness 2: 9, 16, 18, 25, 32, 36, 45, 48, 50, 64, 72, 75, 80, 90, 96, 100; richness 3: 27, 54, 108; richness 4: 81) and 53 are semitotatives (14, 21, 22, 26, 28, …). Of the 53 semitotatives, there are 8 alpharelated semitotatives, 16 omegarelated semitotatives, and 29 opaque semitotatives. In base 120, 11 is a secondorder Wieferich prime. There are 18 centovigesimal quadratic residues (0, 1, 4, 9, 16, 24, 25, 36, 40, 49, 60, 64, 76, 81, 84, 96, 100, and 105). 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. 
Double sharp 
Posted: Nov 7 2017, 11:47 PM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
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.

icarus 
Posted: Nov 8 2017, 04:24 AM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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.

Double sharp 
Posted: Nov 8 2017, 04:49 AM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
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 midscale 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.) 

icarus 
Posted: Nov 8 2017, 12:40 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Double sharp:
The AMT is automated already. It needs several notnecessarily 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: An arbitrary integer x is divisible by …
(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: An arbitrary integer x is divisible by …
(5 glitched. Easy fix.) What can be more indepth than has been done here is a description of the semitotative landscape and left/right trim tricks that are "nonintuitive". 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 alpha2 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 basenaming 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 handwrite tables; that is the big problem. Codegenerated 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 tideover 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). 

Double sharp 
Posted: Nov 8 2017, 02:59 PM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
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 7on18 or 9on14 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! 
Double sharp 
Posted: Nov 18 2017, 04:45 PM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
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 testdrive 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. 
icarus 
Posted: Nov 19 2017, 09:46 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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 numbertheoretical 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. 
Double sharp 
Posted: Nov 19 2017, 11:42 PM

Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
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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