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 Duotrigesimal (base 32), leaving the domain of two
Double sharp
Posted: Jan 31 2017, 09:12 AM


Dozens Disciple


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Joined: 19-September 15



Base 32

Duotrigesimal, SDN: “bioctimal”.

Our exploration of the powers of two brings us to the fifth power, thirty-two. 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 {32} while the omega contributes {7}. The prime 5 remains opaque but the abbreviated square-alpha 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 = 32 * 7 and 65 = 5 * 13, gaining it a magnificent transparency, albeit one trivially equivalent to that of octal if the square-alpha 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 square-alpha test in that 5 and its square are both factors of the square-alpha 1025 = 52 * 41, but whether that is workable is quite questionable when we consider that the thirty-two 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 human-scale range from the high end, and 12 would be even better still, touching down gracefully in its middle.

Letís 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.)

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Double sharp
Posted: Jan 31 2017, 09:26 AM


Dozens Disciple


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Joined: 19-September 15



Octal-Coded Digits of Base 32, using 4-on-8 notation, primes in boldface type

Digit Map Divisibility Tests
  _0 _1 _2 _3 _4 _5 _6 _7
0_ 0 1 2 3 4 5 6 7
1_ 8 9 10 11 12 13 14 15
2_ 16 17 18 19 20 21 22 23
3_ 24 25 26 27 28 29 30 31
  _0 _1 _2 _3 _4 _5 _6 _7
0_ 0 1 2 3 4 5 6 7
1_ 8 9 10 11 12 13 14 15
2_ 16 17 18 19 20 21 22 23
3_ 24 25 26 27 28 29 30 31

Base 32 has the following properties:

  • Decimally, the base is named “duotrigesimal”.
  • Using Systematic Dozenal Nomenclature (see the SDN thread), the base is called “bioctimal”. Read this post describing SDN base names.
  • Thirty-two is a power of two, the prime decomposition of 32 is {2}, which is the set of its distinct prime divisors.
  • Of the 11 primes p less than r = 32, only {2} is a divisor of r. The remaining 10 primes q are coprime to r (i.e., the greatest common divisor (highest common factor) gcd(q, r) = 1).
  • The nth power of a prime will always have n + 1 divisors d | r. The duotrigesimal divisor digits are {0, 1, 2, 4, 8, 16} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 32).) Read this brief description of the divisor.
  • All powers of two r have r/2 totatives t such that gcd(t, r) = 1. Thus 32 has 16 totatives; exactly half the duotrigesimal digits are coprime to r (the totient ratio φ(r)/r = 50…% for all bases r having the single distinct prime divisor {2}). Read this brief definition and description of the significance of the totative.

Regular Digits gr

20 21 22 23 24 25
1 2 4 8 16 32
  • There are 10 neutral digits in the scale of 32. Exactly 31.25% of all duotrigesimal digits are neutral.
  • As a prime power, duotrigesimal cannot have semidivisors (regular numbers gr that do not divide r evenly). The set G of duotrigesimal regular numbers includes only the powers of two. This post defines regular numbers, and their significance.
  • All 10 neutral digits are semitotatives, products pq < r. The set H of semicoprime numbers of base 32 includes any multiple pq with at least one prime divisor p and at least one prime q that is coprime to 32. This includes all even digits that are not integer powers of two. Exactly 31.25% of all duotrigesimal digits are semitotatives.

Positive Primes pr

2 3 5 7 11 13 17 19 23 29 31
  • Let the integers α = (r + 1) and ω = (r − 1). The duotrigesimal α-number is 33, a diprime that distributes alpha benefits to its nonunitary factors {3, 11}. The “alternating-sum” rule that applies to decimal {11} would be applied instead to {3, 11} in base 32. The duotrigesimal ω-totative is 31, a large prime that does not project the omega divisibility rules to any numbers below itself. The “digit-sum” rule would be an obscure number-theoretical curiosity in base 32. See this post for maps and descriptions of the omega and alpha divisibility rules.
  • Duotrigesimal lacks an intuitive divisibility test for the prime 5, however, like the duodecimal test for divisibility by 5, we can use the simple paired-digit alternating sum test to determine divisibility by 5 in base 32, since 32² ≡ −1 (mod 5). This simple rule would prove far less practical in base 32, as there are 204 multiples of 5 to memorize if one wants to ensure recognition of all digit-pairs that are duotrigesimal multiples of 5. This said, this test works for the second power of 5 as well, since 322 + 1 = 1025 = 52 * 41.
  • Seven of the ten duotrigesimal semitotatives and 11 of the 16 duotrigesimal totatives are opaque, thus 18 opaque digits and 14 transparent. Base 32 has an opacity of 56.25%, just over a half.
  • Duotrigesimal reciprocals of primes p < 12 generally enjoy fairly brief expansions. The prime factor {2} terminates after a single place. The omega prime {31} has a single-place reptend. The alpha primes {3, 11} enjoy a two-place reptend. Seven has a three-place repetend and five has a four-place repetend. The larger primes {13, 19, 29} suffer maximal mantissae and {17, 23} are semi-maximal.
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Double sharp
Posted: Jan 31 2017, 09:34 AM


Dozens Disciple


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Joined: 19-September 15



Abbreviated Multiplication Table for r = 32
showing patterns of relationships between products and the base

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32
03 06 09 12 15 18 21 24 27 30
04 08 12 16 20 24 28 32
05 10 15 20 25 30

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 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.

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