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Double sharp 
Posted: Jan 31 2017, 09:12 AM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Base 32Duotrigesimal, SDN: “bioctimal”. Our exploration of the powers of two brings us to the fifth power, thirtytwo. All the powers of two share the same sort of prime signature {n} for prime powers, having n + 1 divisors, and having half of their digits totatives. Their reliance on division into two makes them interesting tools to use that conform quite well to the human tendency towards mediation and duplation, and it is really too bad that numbers don't always work out that way, even if they sometimes have some advantages in their beneficial neighbour relationships conferring some "transparency" for the next few primes that they do not cover. Binary is the smallest base and is nearly entirely exceptional in its properties: it is an exceptionally interesting study but in practice would prove nearly impossible to use due to its lack of compression. Quaternary, grouping binary digits into pairs suffers the same problem. Octal and hexadecimal fall into the middle of the "human scale", being at a happy medium between wanting to split places and wanting to group places: both of them show beneficial neighbour relationships. In hexadecimal, {3, 5} are covered by the omega; in octal, the alpha contributes {3^{2}} while the omega contributes {7}. The prime 5 remains opaque but the abbreviated squarealpha test fills that need, as it does for 13. We may thus expect great things from their sister duotrigesimal, which after all has 6 divisors, as much as the dozen. Unfortunately, our expectations tend to be disappointed regarding this base. The burgeoning of the multiplication tables can mostly be ignored at hexadecimal, but even then we have some issues dealing with the long lines of the four opaque totatives, each sixteen strong: we may need to drill them in using other mnemonics. But in duotrigesimal, we have a total of twelve opaque totatives, and each line is double the length! The mnemonic chores that these would entail are enormous. So it isn't likely that we'd use our current multiplication algorithm in base 32, not to mention the difficulty of memorising all those addition facts. We may of course take advantage of duotrigesimal's binary nature and consider grouping bits in fives, but then the number five does not seem to jive well with its binary composition. Why not group them in fours and use hexadecimal? Perhaps the most robust maintenance of duotrigesimal is as a mixed radix of 4 on 8, though even that is not superior to just using octal itself, and it creates an uncomfortably low concision. The problems are only compounded when we look at extrinsic properties. Octal and hexadecimal both have beneficial neighbours, and even the larger base 64 gains from being flanked by the composites 63 = 3^{2} * 7 and 65 = 5 * 13, gaining it a magnificent transparency, albeit one trivially equivalent to that of octal if the squarealpha test for 5 and 13 is considered. Unfortunately, duotrigesimal has a large prime (31) as its omega neighbour, and while 3 is trivially provided for by the alpha, a semiprime, it unfortunately wastes the last prime rescued by a neighbour relation on 11, a rather large prime that wouldn't seem to find much application. There is a vague glimmer of hope in the squarealpha test in that 5 and its square are both factors of the squarealpha 1025 = 5^{2} * 41, but whether that is workable is quite questionable when we consider that the thirtytwo multiples of 5 up to "50" which would be necessary to memorise to use this test constitute a grave mnemonic load. Even from the point of view of information technology, duotrigesimal strikes out, as while octal had its heyday for encoding bits in triplets, and hexadecimal has now supplanted it by encoding them in quadruplets, there have only very rarely been times when bits need to be grouped into quintuplets. At the scale of base 32, one begins to expect not only divisibility by 2, but also 3 and 5, and duotrigesimal stands as a monument to the fact that it does not just matter how many divisors a number has when considering it as a base, but also which divisors they are, and how far apart they are spaced. As powers of two go, octal or hexadecimal would be a wiser choice; as numbers with six divisors go, 18 or 20 would be wiser choices, concentrating those divisors and entering the humanscale range from the high end, and 12 would be even better still, touching down gracefully in its middle. Let’s take a look at the qualities of 32 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 thread.) 
Double sharp 
Posted: Jan 31 2017, 09:26 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
OctalCoded Digits of Base 32, using 4on8 notation, primes in boldface type
Base 32 has the following properties:
Regular Digits g ≤ r
Positive Primes p ≤ r


Double sharp 
Posted: Jan 31 2017, 09:34 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Abbreviated Multiplication Table for r = 32
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 32, with 57 unique products, about 103.6% the size of the full decimal table and about 73.1% the size of the 12× table commonly learned in grade school. The duotrigesimal AMT is compact and sparsely entrained by regular numbers. The duotrigesimal AMT might be workable and supportive of the complementary divisor method for base 32. This said, the trigesimal AMT is superior to that of base 32. 
