# Prime Factorization Naming Scheme spreadsheet

Obsessive poster
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Joined: Sep 10 2011, 11:27 PM
[d] All numbers decimal

I have put together a spreadsheet in the Cloud, in Google Sheets, at this link: Prime Factorization Naming Scheme. I'm working it up as an aid in exploring the notion of constructing names for whole numbers based on their prime factorizations. The spreadsheet is visible to anyone with the link to it, but not editable. If you would like to experiment with your own ideas, just make a copy of the sheet.  You should automatically get a space in Google Drive if you don't already have one.

In part, my spreadsheet was inspired by the names icarus came up with for his argam numerals (well known to members of this forum), which embody aspects of this notion. However, argam names exhibit some more artful aspects that add to their attractiveness but which make their construction more complex, and therefore harder to automate (such as with a spreadsheet). What I'm aiming for here is something along the same vein, but much more regularized.

This is also more immediately inspired by newcomer Craig's recent thread A Simple and Intuitive Name and Numeral System for Sexagesimal. Even though Craig's goal was limited to providing digits for a positional base no larger than base 60d, his "datraka" naming convention does embed a similar focus on prime factorizations, with prime numbers getting more or less monosyllabic roots that are agglutinated to form the names of composite numbers.  Craig's scheme is limited to giving roots only to the first three primes (2, 3, 5), but of course argam assigns roots for many, many more primes. But I was intrigued by the consonant-vowel (CV) pattern Craig used, where the root for a given prime starts with a given consonant (or consonant cluster), but then the vowel indicates the power to which the prime is raised.

I've been playing around with this kind of CV pattern for a few days, but ultimately I've found it too limiting.  Not enough room to cover a large number of primes in a reasonably intuitive way.  So what I've opted for is something closer to argam:
1. I've given each prime a monosyllabic root name with a CVC pattern, that remains stable for all of its powers. So for example 2 = bin, 3 = trin, 5 = kin, 7 = sep, 11 = lev, 13 = yon, 17 = zot, 19 = wox, etc.
2. I've generally copied icarus's notion of extracting syllables from the name of chemical elements to use as names for corresponding primes. However, I've shaved off trailing vowels or added leading consonants where necessary (usually w-) in order to adhere to the CVC pattern. Hence, zot rather than zote, wox rather than ax, etc.
3. For raising primes to a power, I devised a common set of suffixes with a -VC pattern, where the consonant is a common thematic element, but the vowel indicates the power.  For example ^1 = -im (usually omitted), ^2 = -em, ^3 = -am, ^4 = -om, ^5 = -um.  Note that rather than ordering the vowel sounds alphabetically, I've ordered them by articulation point, from most open and fronted, to most back and rounded.  My impression is that a fronted sound like /i/, with its generally higher frequency pattern, implies "smaller", while a more back sound like /u/, with its generally lower frequency pattern, implies "larger".
4. To get to higher powers than five, I just introduce another thematic consonant and repeat the pattern into another senary place-value: ^6 = -ix, ^12 = -ex, ^18 = -ax, ^24 = -ox, ^30 = -ux.  Combine suffixes for multiple digit powers: -ix = ^6, -ixim = ^7, -ixem = ^8, -ixam = ^9, -ixom = ^10, -ixum = ^11, -ex = ^12, etc.
5. So powers of a prime are formed by appending these suffixes, e.g.: 2 = bin, 4 = binem, 8 = binam, 16 = binom, 32 = binum, 64 = binix, 128 = binixim, 256 = binixem, etc. I've been generally placing the stress accent on the final syllable in such complexes, so that the suffixes won't all degenerate into schwa sounds.
6. Composite numbers are formed by juxtaposing the roots (+ suffixes) of its prime factorization. So for example 6 = trinbin, 10 = kinbin, 12 = trinbinem, 14 = sepbin, 15 = kintrin, 18 = trinembin, 20 = kinbinem, 21 = septrin, 22 = levbin, 24 = trinbinam, 26 = yonbin, 28 = sepbinem, 30 = kintrinbin, 34 = zotbin, 35 = sepkin, 36 = trinembinem, etc.  The final constant/cluster of one prime factor is immediately adjacent to the initial consonant/cluster of the next, so this makes the syllable boundaries between prime factors clear enough.
7. I've opted to order the roots in a composite from largest to smallest. I found that if I did it the opposite way, every other whole number would start with bin, which gets boring; whereas, this way you get more variety up front, yet many numerals wind up rhyming at the end. One could argue that it makes sense to lead with the most significant prime. However, in principle, the order doesn't matter, because multiplication is commutative: 6 = trinbin = bintrin, 12 = trinbinem = binemtrin, etc.
8. Compare argam's 2520 = kinsevoctove, vs. my equivalent sepkintrinembinam.  Icarus orders from smallest to largest factor, bearing in mind that higher powers of even small primes like 2 and 3 can become larger factors.  Whereas here I am ordering from largest down to smallest prime, regardless of power.
9. The primorials Pn are: P1 = 2 = bin, P2 = 6 = trinbin, P3 = 30 = kintrinbin, P4 = 210 = sepkintrinbin, P5 = 2310 = levsepkintrinbin, P6 = 30030 = yonlevsepkintrinbin, etc.
10. I've built the formulas in the spreadsheet so that once you set the name for a given prime, all its occurrences in powers and multiples automatically copy that. Likewise with the list of power suffixes (scroll to the right on the page to find columns defining those). So the spreadsheet can be used to experiment with various choices for syllables.
11. After exhausting all the currently-known chemical elements (up to atomic number 118, oganesson → wog = 647), I went to Presidents of the United States, from George Washington → wash = 653 to Donald Trump → don = 967 (since I already had Harry S. Truman → trum = 877). Then I went to Vice Presidents of the United States (excepting those that were ever President), from Aaron Burr → bur = 971 to Michael Pence → pents = 1193. And I even dug into the Speakers of the House of Representatives of the United States, from Frederick Muhlenberger → muhl = 1201 to Paul Ryan → ryan = 1571, just 3 primes shy of 1600. (Appropriate enough, since the address of the White House is 1600 Pennsylvania Avenue.) I'd like to carry this out at least as far as P5 = 2310, so any suggestions for finding more sources of names would be welcome.  (Hmm, maybe Supreme Court Justices?  Senate majority leaders?  British Prime Ministers? State names? State capitals? ...)
12. This is a naming scheme only. I'm not accompanying it with any notion of how to draw numeral symbols based on their prime factorizations.  I'd defer to something like UTL Numerals, which was the best scheme I've ever seen for doing that. Too bad the anonymous author only posted the images on Facebook for a brief period, until they were taken down suddenly. To my knowledge they have never resurfaced anywhere else.  So all we have left is the image I put together of the first block of 210 (P4) numerals. Whoever this person was, they had filled up the first catalog of 2310 (P5) numerals and looked poised to go on to extend that to 30030 (P6) at least. It was truly remarkable. If anyone knows who the author was, or where UTL has gone now, please let us all know!
So the first block of names is:
 Decimal Dozenal Numeral Name 1 1 un 2 2 bin 3 3 trin 4 4 binem 5 5 kin 6 6 trinbin 7 7 sep 8 8 binam 9 9 trinem 10 A kinbin 11 B lev 12 10 trinbinem 13 11 yon 14 12 sepbin 15 13 kintrin 16 14 binom 17 15 zot 18 16 trinembin 19 17 wox 20 18 kinbinem 21 19 septrin 22 1A levbin 23 1B fluor 24 20 trinbinam 25 21 kinem 26 22 yonbin 27 23 trinam 28 24 sepbinem 29 25 nev 30 26 kintrinbin 31 27 sod 32 28 binum 33 29 levtrin 34 2A zotbin 35 2B sepkin 36 30 trinembinem 37 31 mag 38 32 woxbin 39 33 yontrin 40 34 kinbinam 41 35 lum 42 36 septrinbin 43 37 sil 44 38 levbinem 45 39 kintrinem 46 3A fluorbin 47 3B phos 48 40 trinbinom 49 41 sepem 50 42 kinembin 51 43 zottrin 52 44 yonbinem 53 45 sulf 54 46 trinambin 55 47 levkin 56 48 sepbinam 57 49 woxtrin 58 4A nevbin 59 4B chlor 60 50 kintrinbinem 61 51 gon 62 52 sodbin 63 53 septrinem 64 54 binix 65 55 yonkin 66 56 levtrinbin 67 57 tash 68 58 zotbinem 69 59 fluortrin 70 5A sepkinbin 71 5B cal 72 60 trinembinam 73 61 scan 74 62 magbin 75 63 kinemtrin 76 64 woxbinem 77 65 levsep 78 66 yontrinbin 79 67 teit 80 68 kinbinom 81 69 trinom 82 6A lumbin 83 6B van 84 70 septrinbinem 85 71 zotkin 86 72 silbin 87 73 nevtrin 88 74 levbinam 89 75 chrom 90 76 kintrinembin 91 77 yonsep 92 78 fluorbinem 93 79 sodtrin 94 7A phosbin 95 7B woxkin 96 80 trinbinum 97 81 mang 98 82 sepembin 99 83 levtrinem 100 84 kinembinem 101 85 fer 102 86 zottrinbin 103 87 cob 104 88 yonbinam 105 89 sepkintrin 106 8A sulfbin 107 8B nick 108 90 trinambinem 109 91 cup 110 92 levkinbin 111 93 magtrin 112 94 sepbinom 113 95 zing 114 96 woxtrinbin 115 97 fluorkin 116 98 nevbinem 117 99 yontrinem 118 9A chlorbin 119 9B zotsep 120 A0 kintrinbinam 121 A1 levem 122 A2 gonbin 123 A3 lumtrin 124 A4 sodbinem 125 A5 kinam 126 A6 septrinembin 127 A7 gal 128 A8 binixim 129 A9 siltrin 130 AA yonkinbin 131 AB ger 132 B0 levtrinbinem 133 B1 woxsep 134 B2 tashbin 135 B3 kintrinam 136 B4 zotbinam 137 B5 sen 138 B6 fluortrinbin 139 B7 sel 140 B8 sepkinbinem 141 B9 phostrin 142 BA calbin 143 BB yonlev 144 100 trinembinom 145 101 nevkin 146 102 scanbin 147 103 sepemtrin 148 104 magbinem 149 105 brom 150 106 kinemtrinbin 151 107 kryp 152 108 woxbinam 153 109 zottrinem 154 10A levsepbin 155 10B sodkin 156 110 yontrinbinem 157 111 rud 158 112 teitbin 159 113 sulftrin 160 114 kinbinum 161 115 fluorsep 162 116 trinombin 163 117 stron 164 118 lumbinem 165 119 levkintrin 166 11A vanbin 167 11B yit 168 120 septrinbinam 169 121 yonem 170 122 zotkinbin 171 123 woxtrinem 172 124 silbinem 173 125 zirk 174 126 nevtrinbin 175 127 sepkinem 176 128 levbinom 177 129 chlortrin 178 12A chrombin 179 12B niob 180 130 kintrinembinem 181 131 mol 182 132 yonsepbin 183 133 gontrin 184 134 fluorbinem 185 135 magkin 186 136 sodtrinbin 187 137 zotlev 188 138 phosbinem 189 139 septrinam 190 13A woxkinbin 191 13B teck 192 140 trinbinix 193 141 ruth 194 142 mangbin 195 143 yonkintrin 196 144 sepembinem 197 145 rhod 198 146 levtrinembin 199 147 pal 200 148 kinembinam 201 149 tashtrin 202 14A ferbin 203 14B nevsep 204 150 zottrinbinem 205 151 lumkin 206 152 cobbin 207 153 fluortrinem 208 154 yonbinom 209 155 woxlev 210 156 sepkintrinbin
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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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM
re #11, are you looking for things which have a natural ordering of some kind?

I like this scheme; with only a few exceptions, it doesn't generate words which clash with pre-existing words, making them distinctive.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
SenaryThe12th wrote: re #11, are you looking for things which have a natural ordering of some kind?
Yes.
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)

 Posts 99
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM
Well I guess that rules out supermodels :-)   Names of the Popes or the British monarchs comes to mind, but those are obscure (although if Vice Presidents make the cut, maybe they might work as well).   Romanizations of chinese characters in unicode order would be practically inexhaustable, and be appreciated by a billion people....I wouldn't be one of them *chuckle*.   Books of the Bible, Surah's of the Koran,  Dewey Decimal Classes
https://en.wikipedia.org/wiki/List_of_D ... al_classes

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
SenaryThe12th wrote:Well I guess that rules out supermodels :-)
Miss America pageant winners, on the other hand ...
Names of the Popes or the British monarchs comes to mind, but those are obscure (although if Vice Presidents make the cut, maybe they might work as well).
The problem with those is not so much obscurity as that such persons generally go by single given names, and an ordinal number when the few favorite names get recycled. There would not be enough variety to distinguish one from the other.  In the case of Popes, however, one might use their original pre-election names. In the case of royals, British or otherwise, this wouldn't help because they only came from certain royal houses and so share the same few surnames.
Last edited by Kodegadulo on Sep 7 2018, 03:55 PM, edited 1 time in total.
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)

 Posts 99
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM
Why not just programmatically generate the roots as well?    You could use a simple linear recurrence to generate random-looking syllables.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
SenaryThe12th wrote: Why not just programmatically generate the roots as well?    You could use a simple linear recurrence to generate random-looking syllables.
I might explore that idea, but I think that would actually be better utilized to create a completely different implementation of this concept.  In the implementation you're seeing here, it's just barely possible for someone to memorize the names of a lot of primes, based on vivid associations to chemical elements, presidents, vice presidents, etc. For instance, I've gotten to the point where I can associate 97d with manganese (mang), 101d with iron (fer), 103d with cobalt (cob), etc. But this is by no means the only possible implementation. I'd encourage anyone to try out whatever idea they like.  A more abstract implementation might be more amenable to automatic generation. However, it might be difficult for a human to memorize it, if it lacked any kind of allusion or association they could relate to.

One interesting possibility would be to pick phonemic elements that somehow mapped to the visual elements of UTL numerals, and use those to autogenerate the roots. For instance, UTL distinguishes primes whose modulo in base 6 are 1 vs 5, by using "rounded" shapes for the former, and "angular" shapes for the latter.  I could see that mapped, perhaps, to liquid/nasal consonants, versus plosive stops, respectively.
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)

 Posts 99
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM
If your interests are mnemonic, it might well be easier for somebody to memorize a simple algorithm than to memorize long tables of arbitrary names.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
SenaryThe12th wrote: If your interests are mnemonic, it might well be easier for somebody to memorize a simple algorithm than to memorize long tables of arbitrary names.
That is a plausible hypothesis, and you may very well be right. But people memorize the meanings of thousands of words in their own languages. While some of the grammar and morphology of various languages can be algorithmic, the arbitrariness of root words is anything but. Consider the monumental memorization feat the Chinese must accomplish to learn all the ideograms of their written language. I think you'd be hard pressed to squeeze those into an algorithm. I just have a hunch that it's actually easier to remember something as abstract as a prime integer if it can somehow be associated with a vivid mental image invoked by a name.

But it certainly would be interesting to put it to experiment. If we can devise the equivalent of UTL in phonemes, then we can compare how it scores in memorability against this more argam-like scheme.
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)

 Posts 99
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM
Kodo, it strikes me that if you were to adopt a signed senary representation, perhaps a-la proposal-for-signed-digit-representation-t1909.html, you would get a systematic naming scheme for every rational number which makes its prime factorization explicit.

Obsessive poster
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Joined: Sep 10 2011, 11:27 PM
SenaryThe12th wrote: Kodo, it strikes me that if you were to adopt a signed senary representation, perhaps a-la proposal-for-signed-digit-representation-t1909.html, you would get a systematic naming scheme for every rational number which makes its prime factorization explicit.
You'll need to demonstrate that, because I just don't see it. How can any positional place value representation, in any base, signed or not, possibly make the prime factorizations of all natural numbers self-evident? Any base chosen will necessarily be a finite number, with a finite number of prime factors itself. Only those factors can contribute to what is self-evident in numbers represented in that base. But the potential number of primes that can show up in factorizations is infinite.
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)

 Posts 99
Casual Member
SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM
How can any positional place value representation, in any base, signed or not, possibly make the prime factorizations of all natural numbers self-evident?
Well I wasn't talking about natural numbers, I was talking about rational numbers.  And I didn't say "self-evident", I said "explicit", by which I meant that it made the factorizations explicit in the same way that your proposed schemes makes the factorizations explicit for natural numbers.
You'll need to demonstrate that,
Here's what I was thinking, currently bin <-> two, and trin <-> three.  Therefor, 6 would be trinbin.

For the powers, just use a signed senary number with the following suffexes encoding the digits:

-2 <->  -om
-1 <-> -um
0  <-> -ix
1 <-> im
2 <-> -em
3 <-> -am.

Then you could represent the rational number two-thirds as bintrinum.  You could approximate pi as twenty-two/seven, or levbinseptum, etc etc.  Or something like that.  I'm sure you could come up with a more euphonious scheme than I can.

Obsessive poster
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Joined: Sep 10 2011, 11:27 PM
SenaryThe12th wrote:For the powers, just use a signed senary number with the following suffexes encoding the digits:

-2 <->  -om
-1 <-> -um
0  <-> -ix
1 <-> im
2 <-> -em
3 <-> -am.

Then you could represent the rational number two-thirds as bintrinum.  You could approximate pi as twenty-two/seven, or levbinseptum, etc etc.  Or something like that.  I'm sure you could come up with a more euphonious scheme than I can.
Oh I see. You weren't talking about the roots, you meant the exponents. Sure, we could find a way to represent negative exponents. But I don't think we need to do it by using a balanced signed base. We can just introduce a syllable that means "reciprocal". Let's say, -id:

bin over trin = trinidbin
trin over bin = trinbinid
un over trinbin = trinidbinid
levbin over sep = levsepidbin
binem over trinem = trinemidbinem
trinem over binem = trinembinemid
un over kinbin = kinidbinid
un over kinembinem = kinemidbinemid
un over trinbinem = trinidbinemid
un over trinembinom = trinemidbinomid
kinembinem over trinembinom = kinem over trinembinem = kinemtrinemidbinemid

But perhaps the "X over Y" expressions on the left are already good enough.
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)

 Posts 22
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richard.chasen
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Joined: Apr 28 2017, 04:54 PM
That idea is a spark of genius. I briefly looked it over and I'm impressed.

I have one suggestion that will become important when humans have contact with other intelligent extraterrestial entities. Having a geometric based system for making numbers and then having names and roots based on this geometry. And yes my suggestion would be a difficult project.
I would hardly know where to begin a triangle is the best geometric represention for three, but what would one use for five (or any other number) cubed?