zIFBoards - Free Forum Hosting
Free Forums with no limits on posts or members.

Learn More · Register for Free
Welcome to Dozensonline. We hope you enjoy your visit.
You're currently viewing our forum as a guest. This means you are limited to certain areas of the board and there are some features you can't use. If you join our community, you'll be able to access member-only sections, and use many member-only features such as customizing your profile, and sending personal messages. Registration is simple, fast, and completely free. (You will be asked to confirm your email address before we sign you on.)
Join our community!
If you're already a member please log in to your account to access all of our features:

Name:   Password:


 

 Base 64, The checkerboard base
Double sharp
Posted: Nov 3 2017, 01:49 PM


Dozens Disciple


Group: Members
Posts: 1,402
Member No.: 1,150
Joined: 19-September 15



Base 64

Tetrasexagesimal, SDN: “pentquadral”.

Sixty-four is the sixth power of two; so far, we have examined 8, 16, and 32 among the powers of 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. Each product line of any odd digit through every possible last digit in the multiplication table; apart from those few divisors, which are also the only regular numbers, every even number is a semitotative. It would seem that standing on its own as a pure base, if 16 is borderline human-scale and 32 is already a disappointment, that 64 would seem to offer nothing but a landscape of total desolation for everything except other powers of two.

Furthermore, 64 is the square of 8 and the cube of 4, and hence inherits many of their properties directly. Actually trying to use base 64 seems to devolve instantly into grouping octal places in pairs, since octal is human-scale. Much like bases 100 and 144, it seems that these large squares have no individuality, and that the best way to use base 64 is to not use it at all.

So then why look at base 64 as a base in itself? Like base 144, it carries with it something that lights up the octal world, only more so: as a sixth power, 64 is guaranteed to be showered with beneficial neighbor relationships, even with its poor intrinsic composition! Like any number coprime to 3, it neighbors a multiple of 3 (63); but as a square, it must also neighbor a multiple of 5 (65); as a cube, a multiple of 7 (63); and as a sixth power, a multiple of 13 (65).

These beneficial relationships shower base 64 with a vast number of transparent semicoprimes, and so also octal as its square root. Particularly, octal lacks a direct relationship with 5, but this "square-alpha" test for the number would likely be commonly taught in schools and used often throughout adulthood, perhaps more often than the simpler omega test for 7. The test for 13 would not seem nearly as useful, but being exactly the same, would probably be taught alongside the test for 5.

Letís take a look at the qualities of 64 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.)

Base 64 has the following properties:

Octal-Coded Digits of Base 64, using 8-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
4_ 32 33 34 35 36 37 38 39
5_ 40 41 42 43 44 45 46 47
6_ 48 49 50 51 52 53 54 55
7_ 56 57 58 59 60 61 62 63
  _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
4_ 32 33 34 35 36 37 38 39
5_ 40 41 42 43 44 45 46 47
6_ 48 49 50 51 52 53 54 55
7_ 56 57 58 59 60 61 62 63

Base 64 has the following properties:

  • Decimally, the base is named “tetrasexagesimal”.
  • Using Systematic Dozenal Nomenclature (see the SDN thread), the base is called “pentaquadral”. Read this post describing SDN base names.
  • Sixty-four is a power of two; the prime decomposition of 64 is {2}, which is the set of its distinct prime divisors.
  • Of the 18 primes p less than r = 64, only {2} is a divisor of r. The remaining 17 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 tetrasexagesimal divisor digits are {0, 1, 2, 4, 8, 16, 32} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 64).) Read this brief description of the divisor.
  • All powers of two r have r/2 totatives t such that gcd(t, r) = 1. Thus 64 has 32 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 26
1 2 4 8 16 32 64
  • There are 25 neutral digits in the scale of 64. Exactly 39.0625% of all duotrigesimal digits are neutral.
  • As a prime power, tetrasexagesimal cannot have semidivisors (regular numbers gr that do not divide r evenly). The set G of tetrasexagesimal regular numbers includes only the powers of two. This post defines regular numbers, and their significance.
  • All 25 neutral digits are semitotatives, products pq < r. The set H of semicoprime numbers of base 64 includes any multiple pq with at least one prime divisor p and at least one prime q that is coprime to 64. This includes all even digits that are not integer powers of two. Exactly 39.0625% of all tetrasexagesimal digits are semitotatives.

Positive Primes pr

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
  • Let the integers α = (r + 1) and ω = (r − 1). The tetrasexagesimal α-number is 65, a diprime that distributes alpha benefits to its nonunitary factors {5, 13}. The “alternating-sum” rule that applies to decimal {5} would be applied instead to {5, 13} in base 64. The tetrasexagesimal ω-totative is 63 = 3≤ ∑ 7, distributing omega benefits to its nonunitary factors {3, 7, 9, 21}. Coupled with its native deep power for scanning for 2, all products of any power of two and the prime powers {3≤, 5, 7, 11, 13} are rendered transparent, a range that broadens to include 17 if the paired-digit alternating sum test is included because the square-alpha 4097 = 17 ∑ 241. Base 64 has fortunately positioned neighbors that ameliorate the "one-track mind" common to powers of two. See this post for maps and descriptions of the omega and alpha divisibility rules.
  • Nine of the 25 tetrasexagesimal semitotatives and 20 of the 32 tetrasexagesimal totatives are opaque, thus 29 opaque digits and 35 transparent. Base 64 has an opacity of 45.3125%, just under a half. This further drops to eight opaque semitotatives and eighteen opaque totatives if the square-alpha test is included, reducing the opacity even further to 40.625%, significantly better than base 32.
  • Tetrasexagesimal reciprocals of primes p < 12 generally enjoy fairly brief expansions. The prime factor {2} terminates after a single place. The omega primes {3, 7} have single-place reptends. The alpha primes {5, 13} enjoy a two-place reptend. Nineteen has a three-place repetend and seventeen has a four-place repetend Being a square base, 64 cannot have maximal repetend lengths for any primes: nonetheless, {11, 23, 29} suffer semi-maximal repetends.
Top
Double sharp
Posted: Nov 3 2017, 02:43 PM


Dozens Disciple


Group: Members
Posts: 1,402
Member No.: 1,150
Joined: 19-September 15



Abbreviated Multiplication Table for r = 64
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 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64
03 06 09 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63
04 08 12 16 20 24 28 32 36 40 44 48 52 56 60 64
05 10 15 20 25 30 35 40 45 50 55 60
06 12 18 24 30 36 42 48 54 60
07 14 21 28 35 42 49 56 63
08 16 24 32 40 48 56 64

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 64, with 119 unique products, about 216.4% the size of the full decimal table and about 152.6% the size of the 12× table commonly learned in grade school. The tetrasexagesimal AMT is sparsely entrained by regular numbers, but might be workable and supportive of the complementary divisor method for base 64 in its form as mediation and duplation. This said, the sexagesimal AMT is superior to that of base 64.

Top
« Next Oldest | Large bases (Sixty and up ...) | Next Newest »
DealsFor.me - The best sales, coupons, and discounts for you

Topic Options



Hosted for free by zIFBoards* (Terms of Use: Updated 2/10/2010) | Powered by Invision Power Board v1.3 Final © 2003 IPS, Inc.
Page creation time: 0.0493 seconds · Archive