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Double sharp 
Posted: Nov 3 2017, 01:49 PM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
Base 64Tetrasexagesimal, SDN: “pentquadral”. Sixtyfour 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 humanscale 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 humanscale. 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 "squarealpha" 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. Let’s 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: OctalCoded Digits of Base 64, using 8on8 notation, primes in boldface type
Base 64 has the following properties:
Regular Digits g ≤ r
Positive Primes p ≤ r


Double sharp 
Posted: Nov 3 2017, 02:43 PM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
Abbreviated Multiplication Table for r = 64
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 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. 
