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 Base 54, 3-smooth strength sapped by totatives
Double sharp
Posted: Dec 24 2016, 01:28 PM


Dozens Disciple


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Joined: 19-September 15



Base 54

“Tetraquinquagesimal”, SDN: “quadheximal”.

Base 54 is a mid-scale radix based on two distinct prime divisors: 2 and 3, with its eight divisors {0, 1, 2, 3, 6, 9, 18, 27} making 14.85% of its digits. Six is the smallest quadratfrei composite, and containing within itself the "queen" (2) and "princess" (3) of the primes. As such, its low multiples would all seem to be sensible and good choices for a radix. Base 54 with its two distinct prime divisors is part of the set {12, 18, 24, 36, 48, 54} of 3-smooth multiples of six. With its prime signature {3, 1}, it is also related to the bases {24, 40, 54, 56, 88, 104, 135, 136...} (A065036). Particularly strikingly, the next even number up from 54 is 56, another member of this sequence. Fifty-four however is distinct from all the other listed members of this sequence, in that it is not the lower prime divisor that is cubed, but the higher. This has some consequences for it as a radix. Base 54 retains the same primes {2, 3} in its factorisation as the much lower {6, 12, 18, 24, 36, 48}, but has a very great scale compared to especially the smallest of them, including the eminently human-learnable {12} and the possibly usable {6, 18}. Being two times three cubed, 54 is also related to 24 (three times two cubed), but is over twice the size and still does not include 4 as a divisor, particularly in this neighbourhood when almost all useful bases have it. It seems that, just like its "younger sister" octodecimal, base 54 glorifies ternary powers far too much, robbing us of a better quarter and the ability for everyone to memorise their arithmetic tables.

Base 54 continues to lose steam, much like its double-sized relative 108. It has eight divisors plowing through its multiplication table, but the effect is sparse compared to base 24, which also has eight divisors that make up a third of its digits rather than the 14.85% of base 54. The penetration of the eight divisors of base 54 is not very effective at its scale, compared to that of the twelve divisors of the nearby sexagesimal, base 60. It seems that if we are going to choose a base of prime signature {3, 1}, base 24 is a better deal. Base 12 and perhaps 18 would be even better deals, making efficient arithmetic based on a memorised multiplication table open to all of humanity, instead of the province of only a few mnemonists. Despite its intrinsic weaknesses, base 54 has a glimmer of hope in its neighbour 55, which grants transparency to the important coprime 5 as well as the less important 11. This being said, transparency for 5 can be "bought" for a much lower price with base 24.

It is difficult to imagine a property of 54, with its awkward prime factorisation, that would indicate its use as a civilisational base. Base 54 is surely a mid-scale base, as its huge multiplication table of 1485 distinct values (exactly 27 times the size of the pure decimal table, and about 19 times the size of the 12◊ table sometimes memorized in grade school) makes it too large for human arithmetic when used as a pure number base. We can get around this by using an abbreviated multiplication table, which has 90 distinct values, 115% the size of the aforementioned 12◊ table. If we use a coded approach (alternating bases), we can use nonary (6-on-9) arithmetic, modified for the sub-groups, and not worry about the gargantuan tables. However, due to the paucity of binary divisions of 54, some blunting must be accepted as one of the sub-bases must be odd, subverting the evenness of 54. Generally, a mid-scale base is a composite number base that lies between bases 18 and 30 inclusive, or a runner-up to a highly composite base between 36 and 120 inclusive. Base 54 inhabits the high end of the lower mid-scale, lying between the top of the range of the human scale at 14 (or perhaps 16 or 18) and sexagesimal (base 60).

Letís take a look at the qualities of 54 as a pure number base. Note that this examination is not as thorough as those for smaller bases, for obvious reasons. We use decimal coded figures in this post for your convenience, to alleviate attempting to convert letters and symbols to numeric values.

(Please refer to “Icarus’s Standard Nomenclature for Number Bases” for the legend of the digit map below and any terminology. References to elementary number theory books are given in that post and thread to support what is written in this post.)

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Double sharp
Posted: Oct 22 2017, 08:18 AM


Dozens Disciple


Group: Members
Posts: 1,401
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Joined: 19-September 15



Base 54 has the following properties:

  • Decimally, the base is named “tetraquinquagesimal”.
  • Using Systematic Dozenal Nomenclature (see the SDN thread), the base is called “quadheximal”. Read this post describing SDN base names.

Tetraquinquagesimal Digits

Digit Map Divisibility Tests
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54
  • The prime decomposition of 54 is {2, 3³}, with the set of distinct prime divisors {2, 3}.
  • Of the 16 primes p less than r = 54, only {2, 3} are divisors of r. The remaining 14 primes q are coprime to r (i.e., the greatest common divisor (highest common factor) gcd(q, r) = 1).
  • There are 8 divisor digits d | r: decimal {0, 1, 2, 3, 6, 9, 18, 27} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 54).) Read this brief description of the divisor.
  • There are 18 totatives t such that gcd(t, r) = 1. One third of the digits of base 54 are totatives (the totient ratio φ(r)/r = 1/3 for all bases r having the distinct prime divisors {2, 3}). Read this brief definition and description of the significance of the totative.

Regular Digits gr

  20 21 22 23 24 25
30 1 2 4 8 16 32
31 3 6 12 24 48  
32 9 18 36      
33 27          
  • Base 54 has 8 semidivisors (regular numbers gr that do not divide r evenly): {4, 8, 12, 16, 24, 32, 36, 48}. This set includes any integer 0 < gr that is the product strictly of one or more of {2, 3} that does not divide 54 evenly. Together with the 8 divisors, there are 16 regular digits in base 54; almost three tenths (29.6%) of the digits of base 54 are regular. This post defines regular numbers, and their significance.
  • There are 29 neutral digits in the scale of 54, including the 8 aforementioned semidivisors, the remaining 21 are semitotatives, products pq of at least one distinct prime divisor p and at least one prime q that is coprime to 54. Just over ½ (53.7%) of the digits of base 54 are neutral; the preponderance are semitotatives, making up just under two-fifths (38.9%) of all the digits of base 54.

Positive Primes pr

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
  • Let the integers ω = (r − 1) and α = (r + 1). The ω-totative of base 54 is the large prime 53, which does not share its brief recurrent periods for unit fractions nor the handy indirect intuitive divisibility tests with smaller factors. The digit-sum test would be relegated to curious musings in the back of number theory books in base 54. The alpha neighbour is the semiprime 55, which projects its benefits to the small coprimes {5, 11}. Thus the alpha divisibility rule (the "alternating-sum" rule) would likely find heavy application in a base-54 society, mostly to the prime 5. See this post for maps and descriptions of the alpha divisibility rules for bases less than 30.
  • Base 54 features a decent set of intuitive compound divisibility tests, covering most of the common 5-smooth numbers thanks to the beneficial alpha neighbour. The semitotatives {10, 15, 20, 22, 30, 33, [40], 45} are thus "rescued" from opacity. Base 54 has 26 opaque digits, giving a rather high opacity of 48.1%. Some of the regular tests for “remote” regular numbers like digits {8, 16, 24, 32, 48} prove impractical, which also impacts the compound rule for digit-40. Excluding these gives an even worse opacity of 59.3%.
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Double sharp
Posted: Oct 22 2017, 08:19 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



Abbreviated Multiplication Table for r = 54
showing patterns of relationships between products and the base

1 2 3 4 5 6 7 8 9 a b 10 11 12 13 14 15 16 17 18 19 1a 1b 20 21 22 23
2 4 6 8 a 10 12 14 16 18 1a 20 22 24 26 28 2a 30 32 34 36 38 3a 40 42 44 46
3 6 9 10 13 16 19 20 23 26 29 30 33 36 39 40 43 46
4 8 10 14 18 20 24 28 30 34 38 40 44
5 a 13 18 21 26 2b 34 39 42
6 10 16 20 26 30 36 40 46
7 12 19 24 2b 36 41

The table above is an abbreviated multiplication table or AMT. We can produce such a table by using 1 ≤ nr/2 on one axis and 1 ≤ n ≤ √r for the other, and terminating each product line once the product meets or exceeds r. The table can be minimally abbreviated by considering only unique products, in effect eliminating any repeated product “below” the diagonal line of squares.

The above table shows an abbreviated table for base 54, with 90 unique products, 164% the size of the full decimal table and 127% the size of the 12× table commonly learned in grade school. The base-54 AMT resembles that of bases 36 and 48, only with greater scope, and with different regular products as divisors. Twin totative gaps appear at n = {5, 7}.

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