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Double sharp 
Posted: Nov 4 2017, 09:04 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Base 126“Centohexavigesimal”, SDN: “dechexal”. Base 126 is a high midscale (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, 3^{2}, 7}, relating it to base 90, swapping 7 for 5. Twelve of the base126 digits are divisors; this is however not sufficient to qualify it as a member of the runnersup to the highlycomposite 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 alternatingbase representations might be plausible, particularly 9on14 or 7on18, 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 briefperiod 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…, 1/3^{4} = 0.0,79,0,79…. Thus 3 powers of 5 have reciprocals with minimal recurrent periods, and enjoy the “digitalroot” 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 rangefolding, 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 9on14 or more distantly 7on18, 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 runnerup to a highly composite base between 36 and 120 inclusive. Base 126 may be too great to be a midscale base, but it seems more related to the midscale 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:
Digits of Base 126, using 7on18 or octodecimal coded notation Digit Map
Intuitive Divisibility Tests
Regular Digits g ≤ r
Positive Primes p ≤ r

