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 Base 126, Perils of a too-good neighborhood
Double sharp
Posted: Nov 4 2017, 09:04 AM


Dozens Disciple


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Joined: 19-September 15



Base 126

“Centohexavigesimal”, SDN: “dechexal”.

Base 126 is a high mid-scale (or possibly grand) radix based on three distinct prime divisors, 2, 3, and 7. In this, it is related to bases 42 and 84; its main difference from 84 is its reinforcement of 3 rather than 2. The prime decomposition of 126 is {2, 32, 7}, relating it to base 90, swapping 7 for 5. Twelve of the base-126 digits are divisors; this is however not sufficient to qualify it as a member of the runners-up to the highly-composite numbers (see OEIS A067128), especially since the "great hundred" looms so close - a pattern of failing to live up to its great predecessor in the 120s that continually plagues base 126.

It is fairly difficult to wield base 126; its abbreviated multiplication table is larger than that of 120, which is already likely to be too large for general computation. Several alternating-base representations might be plausible, particularly 9-on-14 or 7-on-18, or vice versa. For a similar composition, it would be better to shift to base 84; for a similar size, the long hundred really seems to be the king of the hill thanks to its comfortable direct and indirect relationships. Thus far, it seems that 126 is outshone too much by the nearby 120: 126 seems to glorify 9 too much, sacrificing a good relationship with the prime 2, lacking the clean quarter that 84 has.

The number 126 has a very interesting neighbor, however, and that is what makes it an interesting stop on the tour. Like decimal 3, 5 is a Wieferich prime to base 126, a coprime prime whose powers share a brief-period recurrent expansion in base r. In fact, 5 is a Wieferich prime to its cube. In base 5, 1/5 = 0.25,25,25…, 1/5² = 0.05,05,05…, 1/5³ = 0.01,01,01…, 1/34 = 0.0,79,0,79…. Thus 3 powers of 5 have reciprocals with minimal recurrent periods, and enjoy the “digital-root” intuitive divisibility tests. The centohexavigesimal world would, despite its pronounced emphasis of the factors 3 and 7 and minimal coverage of 2, possess a deeply penetrating indirect connection to the factor 5. This said, perhaps 126 could have stood to take more care of its own composition; it has only a single binary power, so that already the divisibility test for 4 involves range-folding, and that for 8 is already quite impractical.

Centohexavigesimal seems to be at or past the high end of the “mid scale”, proving too large for human arithmetic when used as a pure number base. Its multiplication table has 8001 unique products, about 145½ times the size of the pure decimal and about 102½ times the size of the 12× table commonly learned in primary school. Even the abbreviated table of 261 unique products, which would allow use of the complementary divisor method, seems excessive: it not only surpasses the table of pure vigesimal, but also is rather broken up. Perhaps the most robust maintenance of base 126 is as a mixed radix of 9-on-14 or more distantly 7-on-18, taming its arithmetic by the smaller tables of tetradecimal or octodecimal. Generally, a mid scale base is a composite number base that lies between bases 18 and 30 inclusive (with 18 and 20 perhaps borderline), or a runner-up to a highly composite base between 36 and 120 inclusive. Base 126 may be too great to be a mid-scale base, but it seems more related to the mid-scale bases 80 and 120 than most other grand bases, having a decent set of direct as well as indirect relationships. This said, although it lives in a rich neighborhood, coupling the great hundred and three perfect powers {121, 125, 128} in a decade, 126 shows up well but not as well as 120.

Let’s take a look at the qualities of 126 as a pure number base. Note that this examination is not as thorough as those for smaller bases, for obvious reasons.

(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.)

Base 126 has the following properties:

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

 

Digits of Base 126, using 7-on-18 or octodecimal coded notation

Digit Map

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g _h
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1_ 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
2_ 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
3_ 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
4_ 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
5_ 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107
6_ 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

Intuitive Divisibility Tests

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g _h
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1_ 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
2_ 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
3_ 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
4_ 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
5_ 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107
6_ 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125
  • The prime decomposition of 126 is {2, 32, 7}, with the set of distinct prime divisors {2, 3, 7}.
  • Of the 30 primes p less than r = 126, only {2, 3, 7} are divisors of r. The remaining 27 primes q are coprime to r (i.e., the greatest common divisor (highest common factor) gcd(q, r) = 1).
  • There are twelve centohexavigesimal divisors d | r: {0, 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 126).) Of all digits of base 126, about 9.52% are divisors. Read this brief description of the divisor.
  • There are 36 centohexavigesimal totatives t such that gcd(t, r) = 1. Just over a quarter of centohexavigesimal digits are totatives (the totient ratio φ(r)/r = 2/7 or about 28.6% for all bases r having the distinct prime divisors {2, 3, 7}). Read this brief definition and description of the significance of the totative.

Regular Digits gr

70 71 72
  20 21 22 23 24 25 26
30 1 2 4 8 16 32 64
31 3 6 12 24 48 96  
32 9 18 36 72      
33 27 54 108        
  20 21 22 23 24
30 7 14 28 56 112
31 21 42 84    
32 63 126      
  20 21
30 49 98
  • Base 126 has 20 semidivisors (regular numbers gr that do not divide r evenly): {4, 8, 12, 16, 24, 27, 28, 32, 36, 48, 49, 54, 56, 64, 72, 84, 96, 98, 108, 112}. Thus there are 32 regular centohexavigesimal digits, including all the divisors and semidivisors, about 25.4% of all digits of base 126. The set G of centohexavigesimal regular numbers includes all positive integer powers and multiples of at least one centohexavigesimal prime divisor p = {2, 3, 7}. This post defines regular numbers, and their significance.
  • There are 72 neutral digits in centohexavigesimal scale, including the 20 semidivisors; the remaining 52 are semitotatives, products pq < r. The set H of centohexavigesimal semicoprime numbers includes any multiple pq with at least one centohexavigesimal prime divisor p and at least one prime q that is coprime to 126. Exactly five-sevenths (57.1%) of the centohexavigesimal digits are neutral, and just over two-fifths (41.2%) of all centohexavigesimal digits are semitotatives.

Positive Primes pr

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113
  • Let the integers α = (r + 1) and ω = (r − 1). The centohexavigesimal α-totative is 127, a large prime that shares none of the omega benefits to smaller digits. The “alternating-sum” rule that applies to decimal {11} would be even more of an academic curiosity in base 126. The centohexavigesimal ω-number is 125 = 53. This number projects the omega divisibility rules to its non-unitary factors, significantly {5, 25, 125}, the “alternating-sum” rule applies to these digits that are powers of 5 in base 125, just as it does to three and nine in decimal. The alpha intuitive divisibility rules would find intensive application to the first three powers of 5, a strong form of centohexavigesimal transparency through intuitive divisibility tests. See this post for maps and descriptions of the omega and alpha divisibility rules.
  • Wendy Krieger’s “sevenites” thread deals with primes q that “divide their own period”, which in standard mathematical literature are called base-r Wieferich primes. As above, base 126 possesses one small Wieferich prime in 53, and none others are known below 144000.
  • Base 126 suffers relatively low digit opacity: 4 of 36 totatives (11.1%) and 16 of 60 semitotatives (26.7%) are unity or involve the transparent factors {5, 5², 5³}. These factors multiply with the relatively small divisors of 126 to produce 16 transparent semitotatives. Intuitive divisibility tests cover 44 centohexavigesimal digits, a 66¼% opacity, although some tests such as those for the remote regular numbers {16, 32, 64} would be impractical. See this post for a divisibility test map of the smallest 30 bases.
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