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Double sharp 
Posted: Dec 24 2016, 01:28 PM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Base 54“Tetraquinquagesimal”, SDN: “quadheximal”. Base 54 is a midscale 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 3smooth 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. Fiftyfour 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 humanlearnable {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 doublesized 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 midscale 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 (6on9) arithmetic, modified for the subgroups, 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 subbases must be odd, subverting the evenness of 54. Generally, a midscale base is a composite number base that lies between bases 18 and 30 inclusive, or a runnerup to a highly composite base between 36 and 120 inclusive. Base 54 inhabits the high end of the lower midscale, lying between the top of the range of the human scale at 14 (or perhaps 16 or 18) and sexagesimal (base 60). Let’s 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.) 
Double sharp 
Posted: Oct 22 2017, 08:18 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Base 54 has the following properties:
Tetraquinquagesimal Digits
Regular Digits g ≤ r
Positive Primes p ≤ r


Double sharp 
Posted: Oct 22 2017, 08:19 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Abbreviated Multiplication Table for r = 54
The table above is an abbreviated multiplication table or AMT. We can produce such a table by using 1 ≤ n ≤ r/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 base54 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}. 
