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 Number Base Theory 101, Some definitions and sources...
icarus
  Posted: Oct 10 2011, 06:34 PM


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Let me begin by disclosing that I am not a professional mathematician nor do I play one on tv. I am a Missouri registered architect and member of the AIA, I am a graphic designer and member of AIGA. I am an avid competitive swimmer and member of the USMS. At most, I am an "amateur mathematician", whatever the heck that means. In recognition of this, let's start with several good sources on number bases and elementary number theory, beginning with the ones in my library:

A.) Ore, Oystein. Number Theory and Its History. Mineola, ny: Dover, 1988. [1st ed. 1948, New York, ny: McGraw-Hill Book Co.]
B.) Dudley, Underwood. Elementary Number Theory. Mineola, ny: Dover, 2008. [2nd ed. 1969, San Francisco, ca: W. H. Freeman & Co.]
C.) Weisstein, Eric W. Wolfram MathWorld. Retrieved October 2011, <http://mathworld.wolfram.com>
D.) LeVeque, William J., Elementary Theory of Numbers. Mineola, ny: Dover, 1990. [1st ed. 1962, Reading, ma: Addison-Wesley Publishing Co.]
E.) Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers. Sixth Edition. New York: Oxford University Press, 2008. isbn 978-0-19-921985-8, 978-0-19-921986-5 (pbk).
F.) Jones, Gareth A. and Jones, J. Mary, Elementary Number Theory. London: Springer (Undergraduate Mathematics Series), 2005.

Now if you're a quick study and ain't got time for the pain, check out Dr. Dudley's book at [B]. Of all the sources, I believe this is the ideal start. [F] is a little more conceptual and textbook-y. [E] starts off okay but gets very deep very soon, and you'll be over your head if you aren't in for a lot of math, but it is a very good book. For mathematicians, this is all child's play, I understand. My old favorite, [A], is a little dated; if you read [B] you're getting everything that's in [A].
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icarus
Posted: Oct 10 2011, 07:13 PM


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Totatives.

Briefly, someone interested in number bases, especially those that facilitate human intuitive computation, will find totatives of interest, since they tend to present resistance to human intuitive computation. There are, however, some aspects of totatives that are beneficial. A composite totative that "neighbors" the number base, e.g., the totative 9 base 10, gives decimal users simple divisibility tests for the totatives 3 and 9, as well as brief recurrent patterns in their decimally-expressed reciprocals. A great deal of totatives would seem to render a number base less "useful" for general intuitive human computation.

Wolfram Mathworld states the following on the totative: "A totative is a positive integer less than or equal to a number n which is also relatively prime to n, where 1 is counted as being relatively prime to all numbers. The number of totatives of n is the value of the [Euler] totient function phi(n). The term was popularized by Sylvester (1879; Dickson 2005, p. 124), who spelled it 'totitive.'" The passage uses the word "popularize" loosely. About as "popular" evidently as "dozenal".

Relatively prime / coprime: "Two integers are relatively prime if they share no common positive factors (divisors) except 1. Using the notation gcd(m, n) to denote the greatest common divisor, two integers m and n are relatively prime if gcd(m, n)=1. Relatively prime integers are sometimes also called strangers or coprime and are denoted m (perpendicular sign) n.

Let's define a "digit". Let the integer r >= 2 be the radix under consideration. Then the digits of r will be all integers 0 < n <= r. The symbol "0" will represent the case where n is congruent to r. Let's ignore the special case where n = 0 (n is actually zero). Check out the definition of digit at Wolfram. I use Wolfram because it is a math resource on the web, and it is not Wikipedia (since some folks don't like Wikipedia).

The digits {1, 5, 7, b} are coprime to one dozen. The digits {1, 3, 7, 9} are coprime to ten. All odd digits are coprime to one dozen four. These are the set of dozenal, decimal, and hexadecimal totatives, respectively. Totatives tend to be a bit resistant to human intuitive computation.

Consider the multiplication table of base r. Let the integer 0 < t < r be coprime to r. This means the gcd(t, r) = 1. In Chicago, they'd say t and r have nothing in common. Let the integer 0 < k be a multiplier. In the multiplication table, the products of totatives have end digits that repeat a pattern every r products as the multiplier k increments. Observe the end digits of the decimal totative 7 {7, 4, 1, 8, 5, 2, 9, 6, 3, 0} and of the duodecimal totative 7 {7, 2, 9, 4, b, 6, 1, 8, 3, a, 5, 0}. This would seem to make the products of a totative t in base r a little more challenging to memorize.

Consider digital expansion of fractions where t is the denominator. The reciprocals of totatives 1/t feature recurrent digital expansions. The decimal fraction 1/7 is 0.142857142857..., the dozenal fraction 1/5 = 0.24972497... This seems to be more inconvenient than the digital expansion of regular numbers, which terminate after one or more places.

Consider intuitive divisibility tests. Totatives generally will not possess an intuitive divisibility test; thus one has to resort to modular math to test for divisibility by seven in base ten, or by five in base twelve.

There are special totatives related to numbers directly neighboring the base which are less resistant. Let the integer omega = base - 1, and let the integer alpha = base + 1. Then there are numbers b coprime to r that divide omega evenly, and there are numbers a coprime to r that divide alpha evenly. (In odd bases, there is a digit 2 which divides both alpha and omega evenly). These neighbor-related totatives offer intuitive divisibility tests and have short recurrent periods in their digital fractions. The digit 3 base ten possesses a divisibility test wherein one adds up the digits of a number and takes a sum; if the sum is itself divisible by three, then the number tested is also divisible by three. In decimal, one third is 0.33333..., and one ninth is 0.111... In hexadecimal, one can test for divisibility by the totatives 3, 5 and digit-15 using the digit sum rule. Thus, the term "opaque totative" applies to any totative that is not the digit 1 nor neighbor-related. Yep I made that term up. I think it is a useful term when we are discussing number bases.

One can count the number of totatives in a given number base by using Euler's totient function. Use EulerPhi[r] in WolframAlpha or Mathematica to calculate the totient function. (I do not know the equivalent in Maple.)

I have a professor friend who thinks I am nuts over totatives because I always talk about them. The fact is I am talking about totatives like the War Department talks about the "Enemy". Lately I have a desire to find composite totatives in convenient places, like right "next to" the number base. Base 120 is interesting, because 119, the "omega", is 7 x 17, and 121 is "alpha", 11^2. This gives the user of base 120 keen divisibility tests for the first five primes plus 17, and brief recurrent digital expansions of the reciprocals of 7, 11, and 17. All of this in a number base that is highly factorable.

Totatives play a big role in cryptography.

Take a look at sources for further information:
[C] Totatives, relatively prime / coprime, [D] page 24, [E] page 58, [F] page 10 for "coprime", [D] pages 11-12 for "alpha" and "omega" in a proof, though the author does not refer to them as "alpha" nor "omega". Divisibility tests for "alpha, omega" [E] pages 142-3, 147. Totient function, see [A] pages 109-13, [B] page 70, [F] pages 86-87.
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icarus
Posted: Oct 10 2011, 07:54 PM


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Divisors.

Briefly, the divisor is an implement that would seem to enable intuitive human computation in a given number base.

Wolfram defines "divisor" thus: "A divisor of a number n is a number d which divides n... also called a factor". This isn't too helpful.

Source [E], Hardy & Wright 2008, Chapter I, The Series of Primes, Section 1.1, "Divisibility of integers", pages 12, specifically: "An integer a is said to be divisible by another integer b, not 0, if there is a third integer c such that a = bc. If a and b are positive, c is necessarily positive. We express the fact that a is divisible by b, or b is a divisor of a, by b | a..."

Let's look at the notion of "divisibility" from source [D]: (LeVeque 1962, Chapter 1.1, "The Euclidean Algorithm and Its Consequences", Section 2-1, "Divisibility", page 22)
"Let a be different from 0 and let b be arbitrary. Then, if there is a c such that b = ac, we say that 'a divides b, or that 'a is a divisor of b', and write a | b ... The following statements are immediate consequences of this definition:
(1) For every a not equal to 0, a|0 and a|a.
(2) If a|b and b|c, then a|c.
(3) If a|b and a|c, then a|(bx + cy) for each x, y. (If a|b and a|c, then a is said to be a common divisor of b and c)
(4) If a|b and b not equal to 0 then |a| less than or equal to |b|."

"Trivial Divisors"
All positive integers r >= 2 possess the divisors {1, r}. These are said to be "trivial" (see source [A] page 29)

Let the integer r >= 2 be the number base, d be a divisor, and d' the divisor complement. Then we can rewrite source [E]'s equation a = bc as r = d * d'. (See source [A] page 86.)

Consider the multiplication table. Let the integer k > 0 be a multiplier. Divisors tend to have brief patterns of end digits in their products, such that the divisor d will have a cycle of end digits that are d' products long. The decimal multiplication table has two nontrivial divisors. The products of 2 end in {0,2, 4, 6, 8} and the products of five in {0, 5}. Duodecimal features four such divisors: 2 {0, 2, 4, 6, 8, a}, 3 {0, 3, 6, 9}, 4 {0, 4, 8}, and 6 {0, 6}. All bases have the trivial divisors, 1 {0, 1, ... } and r {0, 10, ...}. These seem to be easier to memorize than most any other line of products in the multiplication table.

Consider the digital expansion of fractions that have the divisor d as a denominator. All reciprocals of the divisor d in base r will have the expansion 0.d', a single place after the radix point. Very little else can be simpler when it comes to fractions and their digital equivalents. The divisor is a special kind of regular number g, an integer that produces a terminating digital fraction.

Consider the intuitive tests for divisibility by the divisor d. This is a special case of the regular divisibility tests. A regular divisibility test involves examination of the least significant digits of the integer x. In the case of a divisor d, a divisibility test for d is a regular divisibility test that involves comparison with a table of the positive products of d <= r. Thus, divisibility by five in decimal involves looking at the righmost digit of a number, say 95. If the rightmost digit (5) is any of {0, 5}, then the number 95 is divisible by 5. In base twelve, there are more divisors thus more such single-digit divisibility tests. We can test for any of {2, 3, 4, 6} by examining the last digit in base 12.

We can count the number of divisors using the divisor counting function. Read more about it at source [A] page 86 and [B] pages 50-53. Use DivisorSigma[0, r] in Wolfram Alpha or Mathematica. I don't know the Maple function, but it shouldn't be too hard to find. Use Divisors[r] in Alpha or Mathematica to generate a list of divisors.

The divisor is thus a great aid in general human computation. The greater the number of divisors in general, it would seem the greater the number base. The problem is that numbers with more divisors are also large. Since even a handful of distinct prime divisors produces a very large number base, there is a practical limit to how many divisors one can leverage in a number base.

Divisors are a big topic in elementary number theory. Each of the sources has a chapter on such, or on "divisibility."
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icarus
Posted: Oct 10 2011, 08:10 PM


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The Unit.

Let the integer r >= 2 be a number base. The digit 1 is very special. It is neither prime nor composite. It is also both a divisor and a totative of a number base r. This is because 1 | r (one divides r) and the greatest common divisor (highest common factor) gcd(1, r) = 1.
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icarus
Posted: Oct 10 2011, 09:15 PM


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Neutral Digits.

I just wrote a paper on neutral digits. I'll post a link here once it's been reviewed. This is not a "canonical" elementary number theory topic, however the definitions and conjectures are proved in the paper.

Let the integer r >= 2 be a number base. Set d(r) as the number of divisors and t(r) as the number of totatives (I am using t in place of the Greek letter phi, since there is some problem displaying phi right now).

We can produce a "neutral digit counting function" using the following formula:
n(r) = d(r) + t(r) - 1

Note, this formula appears at the OEIS sequence A045763, which is by this same definition, the set of neutral digits. (Edited 12 June 2014.)

We subtract 1 because it is counted both in d(r) and t(r), since it is a divisor of and coprime to r. Observe that many of the composite numbers r > 4 have a significant difference between the number of divisors and non-unitary totatives:
rdtn
221-1
322 0
4320
524 0
642 1
7260
84 4 1
93 6 1
104 4 3
112100
126 4 3
132 12 0
144 6 5
154 8 4
165 8 4
172 16 0
186 6 7
192 18 0
206 8 7

(in the case of r = 2, the difference is -1 since there is only one totative, digit-1.)
In the paper I prove that there are neutral digits for every composite r > 4, and that there are two possible kinds of neutral digit. I've called the two kinds of neutral digits the "semidivisor" and the "semitotative".

Semidivisor.

Let the integer p be a prime divisor of r. A semidivisor is a regular neutral digit that does not divide r evenly, and is the product of at least two prime divisors p.

To define the semidivisor we need the fundamental theorem of arithmetic, and the unique factorization / prime power decomposition formula. See sources [A] page 86, [ B] pages 16-18, [E] page 3, [F] page 20. Let the integer k >= 1, and let the integer p be a distinct prime divisor of r. Let the integer e > 0 be the multiplicity (i.e., the exponent) of p in r.

r = p1e1 p2e2 ... pkek
with p1 < p2 < ... < pk

Then a divisor d of r can be expressed as

d = p1a1 p2a2 ... pkak
with p1 < p2 < ... < pk
and with all akek.

A semidivisor is a regular digit g that has at least one multiplicity ak > ek.
It is too "rich" in at least one prime divisor p, when compared to the base r. Digit 4 base 6 has a prime divisor 2 with multiplicity 2, whilst there is only one 2 in 6. Digit 8 base 10 has a prime divisor 2 with multiplicity 3, whilst there is only one 2 in 10.

I wrote an algorithm in autumn 2013 that can produce the quantity of semidivisors for any given base. This was submitted to the OEIS 11 June 2014 and is now sequence A243822. The semidivisor also appears in an article I wrote for the ACM Inroads in early 2012. Note that a sequence that quantifies regular digits for bases n appears at A010846 (Edited 12 June 2014.)

The semidivisor, like other regular numbers, is generally an asset for those interested in human intuitive computation. In the multiplication table the semidivisor shares the shorter end-digit periods as divisors (there is a relationship between the semidivisor and a complement in these cycles). A semidivisor enjoys a terminating digital expression of its reciprocal. One eighth in decimal is 0.125. One dozen-fourth, a regular number in dozenal is 0;09. The number of places after the radix point will be greater than one, related to the "maximum multiplicity differential" of the semidivisor and the number base. Calculate the maximum multiplicity differential by finding which prime divisor has the highest multiplicity difference with the corresponding prime in the number base. The regular divisibility tests apply to semidivisors. This is a stumbling block for "remote" or "enriched" semidivisors, i.e., those having a high maximum multiplicity differential. Though one can use the regular divisibility test in decimal for eight, this involves knowing the 125 combinations of the three end digits divisible by eight {000, 008, 016, 024, 032, 040, 048, ... 984, 992}.

I've proved in the paper that all composite bases r that are not powers of primes have at least one semidivisor. This rules out base 4 for neutral digits, as it is the square of 2. Base 6 is the smallest base that has a semidivisor, digit-4.

Semitotative.

The semitotative is easier to define. Let the integer q be a prime that is coprime to r. A semitotative is a non-regular neutral digit that is the product of at least one prime divisor p and at least one prime totative q.

In the paper I proved that a given number base, an integer r >= 2 will have at least one semitotative if a minimum totative q is smaller than the complement to the minimum prime divisor p. Thus, every composite number r > 6 will have semitotatives. This leaves r = 4 without neutral digits, and makes r = 6 unique in that it possesses semidivisors, but no semitotatives.

I wrote an algorithm in autumn 2013 that can produce the quantity of semitotatives for any given base. This was submitted to the OEIS 11 June 2014 and is now sequence A243823. The semitotative also appears in an article I wrote for the ACM Inroads in early 2012. (Edited 12 June 2014.)

The semitotative behaves like a semidivisor in the multiplication table, with short end-digit patterns. The digital expansion of multiples of reciprocals of semitotatives is recurrent. Semitotatives as digital reciprocals feature a brief, non-repeating set of digits, then a recurrent mantissa. One sixth in base ten is 0.1666..., one tenth in base twelve is 0;124972497... See source [E] page 142. Like the totative, the semitotative possesses no intuitive divisibility rule unless it is a product of two coprime factors that possess regular or neighbor-related intuitive divisibility tests. This is what I call a "compound intuitive divisibility test" in an upcoming paper.

Thus, with the neutral digits, we have a full "spectrum" of digits with direct relationships to the number base r. This enables the production of digit maps, which indicate the "usefulness" of a number base r, usefulness defined as being beneficial to the intuitive human computation in base r, ignoring the effects of magnitude of r. We can use "digit maps" of number bases r as a shorthand for their utility, since the types of digits speak to their behavior in a few basic applications (multiplication tables, divisibility tests, digital fractions, etc.)
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m1n1f1g
Posted: Oct 10 2011, 09:54 PM


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Can we define semidivisors as digits that do not divide r, but do divide rk for some k > 1? And, if so, can we say how "related" they are to the base by saying the power needed? For example, dozenal 8 would be a semidivisor-2 (8|102), whilst decimal 8 would be a semidivisor-3 (8|103). This is important because the division rules beyond semidivisor-2 are often relatively difficult. Yes, it's the last k digits, but what should they be? I know it's possible, but it's a lot more difficult for semidivisor-3s than for semidivisor-2s.
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icarus
Posted: Oct 10 2011, 10:59 PM


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m1n1f1g,

Yes, that is a plausible definition.

Let's look at an analogous situation. Let the integer p be a prime represented in the prime decomposition of r. Let the integer q be a prime unrepresented in the prime decomposition of r. Then p | r and q is coprime to r.

Using the definition of "coprime" that I normally encounter, we read that an integer a is coprime to b iff gcd(a, b) = 1. To me, that seems more like a "telltale effect". We can say that a number a is a neutral digit of b iff 1 < gcd(a, b) < a. We can also say that a is a divisor of b iff gcd(a, b) = a. A more direct definition would seem to be a is coprime to b if the prime decomposition of b includes no non-unitary divisor of a. (If it isn't "more direct" than perhaps it is a "structural" definition, toward which I am usually biased.)

I'd recommend we use the term "regular number" in general if we are referring to integers that are products powers of the prime divisors p of base r, to the exclusion of any other prime q. Such numbers are restricted to the standard form

n = p1e1 p2e2 ... pkek

This is an existing and well-supported term. Then we have regular digits which are regular numbers less than or equal to r. There are two kinds of regular digits, those that divide r, and those which do not divide r, but as you point out, do divide r^k for some k > 1.

Consider this chart:

|------------------------------------------------| Regular Numbers
|<------Digits----------->|
|----------|-----------------| r
Divisors | Semidivisors

I think you'll find that the distinction between regular digits is less important than portrayed here. My analysis uses r as a dividing line since that is the only line that makes sense. If you want to consider the "remoteness" of the regular digit g from r as a sort of index, I think that is possible. Use this formula

Max[ek - ak]

That is, the maximum difference between the multiplicity a of p in d and multiplicity e of p in r. I suppose we'd need to let any negative value of this differential be truncated to zero, since there is no effect of a "deficient divisor", say digit 6 base 72 (2 * 3 vs. 2^3 * 3 ^ 2) on its behavior across the basic applications. (I have to go to base 72 to get a divisor that fits "under the limbo bar" and isn't a square root; I can't use a prime power, because these have no semidivisor, but they do have regular numbers g > r.)

This would make digit 2 base 10 a regular number at level 0 (a divisor), digit 4 base 10 a regular number at level 1, digit 8 base 10 a regular number at level 2, number 16 base 10 a regular number at level 3, etc. Then we can know the number of places in the terminating decimal representation: the multiplicity differential + 1. This distinction is indeed useful for this purpose. Then the semidivisor is simply a regular number g < r that does not divide r, and a divisor is a regular number whose "maximum multiplicity differential" is zero (or less, if we permit the differential to possess negative value). Thus we have a continuum of regular digits, with special cases. For other purposes, we can acknowledge the special cases, but for the purpose I think you're considering, there isn't really a need to acknowledge them.
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m1n1f1g
Posted: Oct 11 2011, 04:55 PM


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While looking back to your definition of a I found this:
QUOTE (icarus @ Oct 10 2011, 10:15 PM)
Then a divisor d of r can be expressed as

r = p1a1 p2a2 ... pkak
with p1 < p2 < ... < pk
and with all akek.



Shouldn't that be d = p1a1... (not r)?

Anyway, your indices seem better, with divisors being 0 rather than 1. I wanted that, but I couldn't justify it mathematically.

I thought "regular numbers" were numbers which are 5-smooth. They're effectively what you would call regular numbers of base 26.
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icarus
Posted: Oct 11 2011, 06:30 PM


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M1n1f1g,

I've fixed it. Normally I don't "edit" messages but with these posts I will do so, so that they are correct. Later this will go on the DSA website as part of a larger effort, probably in early 2012.

Regular Number.
The notion of "regular number" does arise in the study of Mesopotamian Records (Duncan Melville). This Wikipedia definition of regular number uses this definition. Let the integers {a, b, c} >= 0 be the exponents of the prime divisors 2, 3, and 5, respectively. Let n be an integer. A regular [sexagesimal] number is a number of the form
n = 2a 3b 5c
This is the Sloane number sequence A051037, there called "5-smooth numbers: i.e. numbers whose prime divisors are all <= 5":
{1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, ...}
The integer multiples of reciprocals of regular [sexagesimal] numbers enjoy a terminating [sexagesimal] digital expansion. A comment at Sloane's OEIS.org reads, "Sometimes called the Hamming sequence, since Hamming asked for an efficient algorithm to generate the list, in ascending order, of all numbers of the form 2^i 3^j 5^k for i,j,k >= 0. The problem was popularized by Edsger Dijkstra." This is the definition at Wikipedia and equivalent to the formula in this post, immediately above. This definition of "regular [sexagesimal] number" is thus equivalent to "5-smooth number". You would be correct in interpreting a "5-smooth number" also to be a regular number in base thirty, since the distinct prime divisors of thirty {2, 3, 5} are those of sixty {2, 3, 5}. The 5-smooth numbers are also regular in bases 120 and 360, and are a subset of the regular numbers of base 2520, since the distinct prime divisors of the first two are {2, 3, 5}, but those of 2520 are {2, 3, 5, 7}.

The definition of regular number at MathWorld talks of a decimal regular number. Using the variables we set immediately above, a regular [decimal] number is a number of the form
n = 2a 5c
This is the Sloane number sequence A003592, simply referred to as "Numbers of the form 2^i*5^j", avoiding the term "regular [decimal] numbers":
{1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, ...}
Nonetheless, a description reads, "These are the natural numbers whose reciprocals are terminating decimals". This is equivalent to the definition at MathWorld and the formula in this post immediately above. Examples in source [E] starting at page 140 use the regular [decimal] number for exploration of representation of numbers by decimals. (See Hardy & Wright 2008, Chapter IX, "The Representation of Numbers by Decimals".) The same number sequence are regular vigesimal numbers, since the decimal distinct prime divisors {2, 5} are also the vigesimal distinct prime divisors {2, 5}.

Source [A] page 316 states the following:
Ore 1948, Chapter 13, "Theory of Decimal Expansions", page 316, specifically: "In general, let us say that a number is regular with respect to some base number b when it can be expanded in the corresponding number system with a finite number of negative powers of b. ... one concludes that the regular numbers are the fractions r = p/q, where q contains no other prime factors other than those that divide b."
This is a more general definition than those at MathWorld/Melville and Wikipedia/Hardy & Wright. Clearly the definitions at MathWorld/Melville and Wikipedia/Hardy & Wright are simply base-specific definitions of a larger concept of "regularity", i.e., those numbers g such that the prime divisors p of g are limited to those distinct prime divisors p found in the prime decomposition of the number base r, and exclude all prime numbers q not found in the prime decomposition of r.

The definition I use is the general definition of regular number given by Ore, by analogy of using "digital fraction" in place of "decimal fraction", "radix point" or "unit point" in place of "decimal [point]", etc. I use "5-smooth number" or "regular sexagesimal number" to refer to numbers of the form
n = 2a 3b 5c
and "regular decimal number" for those of the form
n = 2a 5c.

The regular dozenal numbers, {1, 2, 3, 4, 6, 8, 9, 10;, 14;, 16;, 20;, 23;, 28;, ...} I refer to as "three-smooth" or "regular dozenal" numbers. Using variables set at the start of this post, these are of the form
n = 2a 3b.
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dgoodmaniii
Posted: Oct 12 2011, 06:41 PM


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Keep them coming, please! I'm learning more number theory in this thread than I did in fourteen years of public schooling.
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m1n1f1g
Posted: Oct 12 2011, 10:01 PM


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QUOTE (dgoodmaniii @ Oct 12 2011, 07:41 PM)
Keep them coming, please!  I'm learning more number theory in this thread than I did in fourteen years of public schooling.

Good observation. The only one of these terms I've learnt in school is "divisor" (well, "factor" as it is often called). I haven't done A levels yet, but I can't see them cropping up there. The concepts are interesting to study, but probably only by people who are already interested in maths. I suppose they're a bit difficult to test on, beyond "How many totatives does the number 16 have?". There should be a coursework task to write about something interesting in an interesting way, I'm sick of writing rushed history essays for exams. angry.gif

On the regular numbers thing, according to your evidence you seem entitled to call them regular numbers for an arbitrary base. The evidence fits logically in an OR operation of proof, meaning that you've already proved your point. Actually, that really meant nothing, all I'm saying is that you can continue as before.
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icarus
Posted: Oct 13 2011, 03:30 PM


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Euler Totient Function.

m1n1f1g implies above that the question of "how many totatives does the number 16 have" is perhaps a trivial question, and in everyday life it might be so. Another way to state "how many totatives does the number 16 have" is "how many numbers 0 < n <= r are coprime to r = 16?" This question is a central topic in elementary number theory, studied by the greatest mathematicians, Euler, Fermat, principally. The following sources have chapters on the function:

[ A ] Chapter 5.5 "The Aliquot Parts", "Euler's Function"
[ B ] Section 9 "Euler's Theorem and Function"
[ D ] Chapter 3.4 "Reduced Residue Systems and Euler's [phi] Function"
[ E ] Chapter 5.5 "Congruences and Residues", "Euler's Function phi(m)"
[ F ] Chapter 5 "Euler's Function"

The Euler function, for the rest of the world outside of those who love number bases, is more closely associated with the notion of congruences / modular math and the greatest common divisor (highest common factor) than "totatives". The function counts how many numbers 0 < n <= r have gcd(n, r) = 1. (Remember that gcd(n, r) = 1 is usually cited as the definition for "coprime / relatively prime", it is the sure test that indicates n and r are coprime.)

This page summarizes the RSA public-key cryptography algorithm. (Read more about RSA at Wikipedia) If you visit it, you'll see that the Euler totient function is highly important to the algorithm. In fact, if you are planning to encrypt a message, you want a lot of "resistance", you want as many totatives as you can get. If you're encrypting a message, you need large prime numbers. The document provides references and does a good job of explaining the basics of the method. This method was developed (looking at Wikipedia) in the seventies. This is an example of how a seemingly unimportant basic mathematical property can become a seriously powerful application. Imagine what today's internet/computer society would be like without public-key encryption?

Because the Euler totient function was discovered around the time when modular math, congruences, and residues were, strangely this end of the "digit spectrum" is well-founded in mathematics. Thankfully we can "detect" how many totatives there are in a base, how many integers 0 < n <= r are coprime to r without having to count them like this example (but the row examples are a neat way of studying the function).

Let the integer r >= 2 be a number base. Let the prime p be a prime divisor of r (i.e., p | r). Let the integer e > 0 be the multiplicity (i.e., exponent) of p in r. Let the integer k >= 1. Consider the standard prime power decomposition of r:

r = p1e1 p2e2 ... pkek
with p1 < p1 < ... < pk

Sources [A, p. 110], [C, formula 1], [D, pp. 43-44] state that

φ(p) = p - 1.

This means every prime has (prime - 1) totatives. Using the nomenclature earlier established in this thread

Let ω = p - 1;
φ(p) = ω.

Reading further in these sources, you'll see the function is multiplicative, dependent on the distinct prime divisors pk of r.
Sources [A, p. 113], [B, p. 70], [C, formulae 12-13], [D, p. 45], [E, pp. 64-65], [F, p. 87] show that we can compute the Euler totient function by

φ(r) = r (1 - 1/p1)(1 - 1/p2) ... (1 - 1/pk)

Please do not neglect to check the sources. These sources have more information about any of the topics than can be posted on this forum. I think anyone interested in number bases should be familiar with the things that have been posted thus far (except perhaps the "neutral digits" as presented here; folks should know about regular digits, though. I think the neutral digits paper fills in a gap and allows us to succinctly analyze any number base r). Many of the sources ([A], [B], [D]) are available from Dover, meaning that they aren't that expensive. Source [C], MathWorld, is a good internet reference, like an encyclopedia for maths. Hardy & Wright (source [E]) and Jones & Jones (source [F]) are more expensive and more serious. There are things I doubt I'll ever understand in both of them.

One never knows where this knowledge will take you. I tutor eighth grade (13-14 year old for the UK readers on this site) girls in pre-algebra. Many many times the comment pops up "why do we even have to know this," "if it can't equal six than why do they even write a six, why don't they write seven," etc. We live in a world where algorithms increasingly determine outcomes, where one can make billions on an algorithm. Number theory, mathematics, and logic are the keys to the algorithms.
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daedalus
Posted: Oct 14 2011, 05:28 PM


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m1n1f1g
Posted: Oct 14 2011, 10:25 PM


Dozens Disciple


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Were you supposed to be logged in as daedalus? I thought that was your quick-fix temporary profile.

Anyway, it's surprising to see how many bases are better than binary in this respect, but also how that side is far slower to reach 0 than the primes are to reach 1. It's common sense really, that when r = infinity, r/(r-1) = 1; but it's amazing to think that this base may have no totatives, if it is a multiple of everything. Not that that'd be useful. Every number would be a special case, and each new multiplication attempted would need to be worked out manually, by adding numbers repetitively and looking them up on some colossal number line. But that's stupid tongue.gif.
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icarus
Posted: Oct 14 2011, 10:43 PM


Dozens Demigod


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Yes, I've noticed that. That was a mistake. When I used this device I had an autofill and it put in what I logged in last time. It almost happened twice!

When you say "better than binary" you mean "better than the number bases that are powers of two" and can include any number bases that are powers of any single prime. "Better" here meaning having a lower proportion of totatives, which represent resistance to human intuitive computation.

I think binary is the best base, if you are not limited by human cognitive ability. Specifically, if you can parse "11111011011" as fast as "2011" decimal. This is what I can't study: what is the lowest base that does not throw us for a loop when we try to parse strings of digits? To me, it seems like 7 or 8. I have a way to gauge it but it isn't as steeped in mathematics as the facts conspire against large bases.
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icarus
Posted: Oct 14 2011, 10:45 PM


Dozens Demigod


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The Digit Map.

Given everything thus far in this thread, we can construct a map the digits of each number base that shows the relationship of a digit n to the number base n. The digit maps offer a shorthand way of evaluating the behaviour of each digit of the bases mapped.

      Digit Maps for Bases 2 ≤ r ≤ 26;
2 0 1                                   Key  
3 0 1 2                                 Unit, 1 | r ∧ 1 r  
4 0 1 2 3                               Divisor, d | r  
5 0 1 2 3 4                             Semidivisor, g = piδ ∧ (δ > ρ) ∧ (g ¬| r) ∧ (0 < gr)  
6 0 1 2 3 4 5                           Semitotative, h = piqj ∧ (0 < h < r)  
7 0 1 2 3 4 5 6                         Totative (opaque), t r ∧ 0 < t < r  
8 0 1 2 3 4 5 6 7                       α-Totative, tα r ∧ , tα | (r + 1)  
9 0 1 2 3 4 5 6 7 8                     ω-Totative, tω r ∧ , tω | (r − 1)  
X 0 1 2 3 4 5 6 7 8 9                   α/ω-Totative, tαtω  
E 0 1 2 3 4 5 6 7 8 9 X                                        
10 0 1 2 3 4 5 6 7 8 9 X E                                      
11 0 1 2 3 4 5 6 7 8 9 X E 10                                    
12 0 1 2 3 4 5 6 7 8 9 X E 10 11                                  
13 0 1 2 3 4 5 6 7 8 9 X E 10 11 12                                
14 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13                              
15 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14                            
16 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15                          
17 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16                        
18 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17                      
19 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18                    
1X 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19                  
1E 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X                
20 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E              
21 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20            
22 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21          
23 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22        
24 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22 23      
25 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22 23 24    
26 0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22 23 24 25  
  0 1 2 3 4 5 6 7 8 9 X E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22 23 24 25 26

Visit this link for a PDF with digit maps for all bases 2 <= r <= 120.
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icarus
Posted: Oct 14 2011, 10:49 PM


Dozens Demigod


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It may be easier to see the behaviour of the different types of digits in a multiplication table using the following diagram. The colour scheme is the same as defined in the above digit map.


user posted image
This is part of a larger work that applies the analysis to all number bases below 26; that will be available at the DSA website. There is another method of analysis that is planned, but I am far from finishing it.
Ciao for now, family dinner, a patriarch is having his birthday gala and I get to sit with all the kiddies!
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icarus
Posted: Dec 5 2011, 04:17 PM


Dozens Demigod


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Icarus’ Standard Nomenclature for Number Bases.

This message is posted in order to provide a brief one-stop reference for many of the terms I use in other posts. It is assumed that you are familiar with the following concepts:

  • Definition of an integer
  • Primality or composition (primes, composites, and the integer 1 as neither prime nor composite, called a “unit”) [1, 2, 3]
  • Definitions and concepts of a number base, positional notation

Number Base (Radix). Let the integer r ≥ 2 be a number base (or radix), and let the integers b and n be greater than or equal to 0. We can represent an arbitrary integer a as

[1]     a = br + n     [4]

Digit. Let n be a digit of base r, with 0 < nr. The digit “0” is used to signify that an arbitrary integer a is congruent with r, i.e., a ≡ 0 (mod r) [5]. We will ignore the case where a = 0. There must be r numerals to represent base r in standard positional notation. [6] In general (since the start of 2011), I use the term “digit” to mean an integer n, “numeral” to mean the symbol and name of n, and “place-value” to signify a digit in a number such as “2011”, wherein the most significant place-value is 2. (See Wolfram MathWorld’s definitions of Digit and Numeral).

Prime Decomposition. Each positive nonzero integer can be uniquely represented as the product of powers of primes [7]. Let the prime p be a divisor of r, i.e. p | r, and let the prime q be coprime to r, i.e. qr. We can write the standard form for prime power decomposition of r as:

[2]     r = p1ρ1 · p2ρ2 · … · pkρk
with p1 < p2 < … pk

Unit. A unit is the digit 1, neither prime nor composite, both a divisor of and coprime to r. Units appear in purple in the digit map below.

Figure 1: Decimal Unit Digits

0 1 2 3 4 5 6 7 8 9

Divisor. A divisor is an integer d that, multiplied by a second integer d′, produces r [8]:

[3]     r = d · d′.

Let the integer k ≥ 1 represent the number of distinct prime divisors p of r. Let the integers ρ > 0 and δ ≥ 0 be the exponents of any prime factor of the number base r and its divisors d, respectively. The divisor d of r has the prime decomposition

[4]     d = p1δ1 · p2δ2 · … · pkδk
with p1 < p2 < … pk and with no δ > ρ

Thus, no divisor of the number base can have an exponent of any prime factor exceed that of the corresponding prime factor in the number base itself. The divisor digits {0, 1} (the divisors {1, r}) are divisors of each base r, and are called “trivial divisors” [9]. The divisor is a kind of regular number. See the MathWorld definition of divisor. See OEIS A027750 for divisors of r and the divisor counting function OEIS A000005. Divisors appear in red (purple for the unit divisor) below.

Figure 2: Decimal Divisor Digits

0 1 2 3 4 5 6 7 8 9

Regular number. Let the integer g have the form shown in Formula 4, without the restriction that “no δ > ρ[10, 11]. The regular number g thus is a product solely of prime divisors p of base r, with any exponent δ > 0. The decimal regular numbers include products g of any positive power of 2 and any positive power of 5: {1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, …}.

Regular digit. A regular digit is a regular number gr, with the digit “0” signifying congruence with r. The decimal regular digits are {0, 1, 2, 4, 5, 8} whilst the regular numbers less than or equal to r are {1, 2, 4, 5, 8, 10}. Thus the set of divisor digits is a subset of the set of regular digits of r. Regular digits appear in red, purple, and orange below. See OEIS A162306 for regular digits of base r and the “regular digit counting function” OEIS A010846.

Figure 3: Decimal Regular Digits

0 1 2 3 4 5 6 7 8 9

“Semidivisor”. I call a composite nondivisor regular digit g < r a “semidivisor”. The decimal semidivisors are {4, 8}. Semidivisors are one of two types of “neutral digit”. Semidivisors must be composite digits of composite bases r. Semidivisors appear in orange below. (Another term I considered was “quasidivisor”, but selected “semidivisor” in symphony with the term “semitotative”, the other kind of neutral digit.) See OEIS A272618 for semidivisors of r and the “semidivisor counting function” OEIS A243822.

Figure 4: Decimal Semidivisors

0 1 2 3 4 5 6 7 8 9

Semidivisors exist for all composite number bases r ≥ 6 that are not perfect prime powers.

Totative. Let the integer t < r be such that gcd(t, r) = 1; thus t is coprime to base r, i.e., tr [12, 13, 14]. Then t has the prime decomposition

[5]     t = q1ρ1 · q2ρ2 · … · qkρk
with q1 < q2 < … qk

The totative t and the number base r have no factor in common but 1; t is “out of phase” with r. See the MathWorld definitions of relatively prime and totative. The set of totatives of r is also known as “reduced residue system” of r. See the Euler totient function OEIS A000010, and A038566 for the reduced residue system. Totatives appear in gray, light blue, and purple (for the unit totative) below. They are colored light green in Figures 6 and 7.

Figure 5: Decimal Totatives

0 1 2 3 4 5 6 7 8 9

Neighbor-Related Totative. Let the integers α = (r + 1) and ω = (r − 1). A neighbor related totative is a divisor of either α or ω [15, 16]. The digit 2 in odd bases r is a divisor of both α or ω. The neighbor-related totatives cannot divide r, since 2 is the smallest prime, and the difference between r and r ± 1 is by definition less than 2.

Figure 5: Decimal Neighbor-Related Totatives

0 1 2 3 4 5 6 7 8 9

Figure 6: Neighbor-Related Totatives in Base Fourteen

0 1 2 3 4 5 6 7 8 9 a b c d

Figure 7: Neighbor-Related Totatives in Base Fifteen

0 1 2 3 4 5 6 7 8 9 a b c d e

Alpha-Related Totative (sometimes called “alphas” though technically only r + 1 = α). A divisor of α. Let tα be a divisor of (r + 1). Then tα cannot divide r evenly, since no prime is smaller than 2. All the prime factors of (r + 1) must be q, thus gcd(r, (r + 1)) = 1. Alpha-related totatives appear in light-green in Figures 6 and 7 above, and in light-purple (digit 2 base 15) in Figure 7 above.

Omega-Related Totative (sometimes called “omegas”) though tecnically only r − 1 = ω. A divisor of ω. Let tω be a divisor of (r − 1). Then tω cannot divide r evenly, since no prime is smaller than 2. All the prime factors of (r − 1) must be q, thus gcd(r, (r − 1)) = 1. Omega-related totatives appear in light-blue in Figures 5–7 above, and in light-purple (digit 2 base 15) in Figure 7 above.

Alpha/Omega-Related Totative For all odd bases r, digit 2 | (r − 1) and 2 | (r + 1). Thus, digit 2 in all odd bases r is related to both α and ω. The behavior of digit 2 in odd bases is that of an ω-related totative. The alpha/omega-related totative 2 appears in light purple in Figure 7 above.

“Opaque” Totative. I call a totative “opaque” if it is a digit tr that does not divide either or both α = (r + 1) nor ω = (r − 1). The notion of “opacity” relates to the notion of totatives as providing little leverage for human intuitive computation. Some opaque totatives are not “opaque“ concerning divisibility tests. Totatives that inherit intuitive divisibility tests from an α- and ω-related totatives indeed have composite intuitive divisibility rules.

Figure 8: The Decimal Opaque Totative

0 1 2 3 4 5 6 7 8 9

“Neutral Digit”. I call a digit that is neither a divisor nor a totative of r a “neutral digit”; these are nondivisors in the cototient of r. Neutral digits must be composite digits. There are two types of neutral digit: the semidivisor (see above), and the semitotative. The semidivisor is a product solely of any power δ of any of the prime divisors p of base r, and is a regular number g le; r. The semitotative is a mixed product pq, with p a prime divisor of r and q a prime that is coprime to r. The semitotative is a semi-coprime integer h < r. Neutral digits appear in gold in Figure 9. See the “neutral digit counting function” OEIS A045763 and a list of neutral numbers less than r at A133995. See also a set of proofs I wrote about neutral numbers here.

Figure 9: Decimal Neutral Digits

0 1 2 3 4 5 6 7 8 9

“Semi-Coprime Number”. A “semi-coprime number” is an integer h that is simply the product pq of at least one prime divisor p and at least one prime q that is coprime to r [17]. The decimal semi-coprime numbers are {6, 12, 14, 15, 18, 21, 22, 24, 26, 28, 30, …}. The semitotative relationship to semi-coprime numbers is the same as the semidivisor relationship to regular numbers: these are simply the portion of the larger set that is less than or equal to r, thus able to be represented as digits of r.

“Semitotative”. Consider a composite digit h < r that is the product pq of at least one prime divisor p and at least one prime totative q of r. I call such a digit a “semitotative”. Semitotatives exist for all composite bases r that are not powers of primes p (rpρ). Thus octal is the smallest r to possess a semitotative (digit 6). For large, highly composite bases r, semitotatives abound, comprising the predominance of digits of r, but semitotatives are rare in bases 2 ≤ r≤ 20. The set of semitotatives of r are a subset of the semi-coprime numbers h. The decimal semitotative is digit 6. The hexadecimal semitotatives are digits {6, 10, 12, 14}.

“Opaque” Semitotative / Semi-Coprime Number. I call a semitotative or semi-coprime number “opaque” if it is the product pq with no intuitive divisibility test for q in base r. See OEIS A272619 for semitotatives of r and the “semitotative counting function” OEIS A243823.

Figure 10: The Decimal Semitotative

0 1 2 3 4 5 6 7 8 9

“Digit Map”. Given the types of digits described above, it is possible to produce a “map” of all the relationsips of digits n with base r. See Figure 11. This uses the legend in Figure 12.

Figure 11: The Decimal Digit Map

0 1 2 3 4 5 6 7 8 9

Figure 12: Legend

Unit, 1 | r ∧ 1 ⊥ r

Divisor, d | r

Semidivisor, g = ∏ piδ ∧ (δ > ρ) ∧ (g ¬| r) ∧ (0 < gr)

Semitotative, h = ∏ piqj ∧ (0 < h < r)

Totative (opaque), tr ∧ 0 < t < r

α-Totative, tαr ∧ , tα | (r + 1)

ω-Totative, tωr ∧ , tω | (r − 1)

α/ω-Totative, tαtω

“Digit Spectrum”. We can produce a chart similar to the digit map wherein we are simply representing the ratio of the quantity of digits of a given type to the magnitude of r. This is especially useful for large bases r.

Figure 13: The Decimal Digit Spectrum

                   

References:

See the OP for the list of books referenced in the following endnotes:

  1. Dudley 1969, Section 2, “Unique factorization”, page 10.
  2. LeVeque 1962, Chapter 1, “Foundation”, Section 1�C5, “Radix representation”, pages 17–19.
  3. Hardy & Wright 2008, Chapter I, “The Series of Primes”, Section 1.2, “Prime numbers”, page 2.
  4. Jones & Jones 2005, Chapter 2, “Prime Numbers“, Section 2.1, “Prime numbers and prime-power factorisations“, page 20.
  5. Dudley 1969, Section 4, “Congruences”, page 28.
  6. Dudley 1969, Section 13, “Number in Other Bases”, page 106.
  7. Dudley 1969, Section 2, “Unique factorization”, pages 16�C18
  8. Ore 1948, Chapter 5, “The aliquot parts”, page 86.
  9. Ore 1948, Chapter 2, “Properties of numbers. Division”, page 29.
  10. Ore 1948, Chapter 13, “Theory of Decimal Expansions”, page 316.
  11. Hardy & Wright 2008, Chapter IX, “The Representation of Numbers by Decimals”, Section 9.2, “Terminating and recurring decimals”, page 142.
  12. Hardy & Wright 2008, Chapter V, “Congruences and Residues”, Section 5.1, “Highest common divisor and least common multiple”, page 58.
  13. Jones & Jones 2005, Chapter 1, “Divisibility”, Section 1.1, “Divisors”, page 10.
  14. LeVeque 1962, Chapter 2, “The Euclidean Algorithm and Its Consequences”, Section 2�C2, “The Euclidean algorithm and greatest common divisor”, page 24.
  15. LeVeque 1962, Chapter 1, “Foundation”, Section 1�C3, “Proofs by Induction“, pages 11�C12.
  16. Hardy & Wright 2008, Chapter IX, “The Representation of Numbers by Decimals”, Section 9.6, “Tests for divisibility”, pages 142-143, 147.
  17. Hardy & Wright 2008, Chapter IX, “The Representation of Numbers by Decimals”, Section 9.2, “Terminating and recurring decimals”, pages 142-145.
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icarus
Posted: Apr 26 2012, 10:43 PM


Dozens Demigod


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Digit-wise relationships with Number Bases

A running project of mine is to build a truly massive digit map of all bases up to 2520 for a really huge poster I plan to hang in my office. In doing this I’ve found it is easier to build if I proceed “digit-wise” rather than base-wise.

Let the integer r ≥ 2 be a number base (or radix). Let n be a digit of base r, with 0 < nr. The digit “0” is signifies that an arbitrary integer a is congruent with r, i.e., a ≡ 0 (mod r). We will ignore the case where a = 0. In much of my work with digit maps, we have examined the base-wise relationship of n to r. This involves looking at the (horizontal) rows of digits n of a number base r. We can also examine the simpler relationship of bases r to digits n by examining the (vertical) columns. Unlike the base-wise, horizontal data, the digit-wise data are cyclical. We can regard r as a residue ρ (mod n).

rρ (mod n)

When we look at the number bases r through the lens of the digits n, we can summarize the relationship by only examining the single cycle using modular math. Thus instead of looking directly at the number base r, we look at its residue ρ (mod n). Also, examining the digit maps digit-wise, we can more easily construct such maps as we can completely avoid piecewise factoring. All we need to know are the prime decompositions of ρ and n.

We can construct a digit-wise digit map by laying out a space for each residue ρ of the integer n. We’ll lay out the map horizontally here for convenience, though in the digit maps commonly posted here, such data occurs vertically. We’ll use the same color convention to indicate divisors, semidivisors (nondivisor regular numbers), semitotatives (semi-coprime numbers), totatives, and neighbor-related totatives. From the point of view of digits, these colors show the same relationships but have meanings relative to digits rather than to bases.

Case 1: gcd(ρ, n) = 1. This case describes a residue ρ coprime to n. This case will be colored light gray.

Case 2:  r ≡ ±1 (mod n). Since |ρ| = 1 and gcd(1, n) = 1, r and n are coprime. Let’s look at the sub-case r ≡ +1 (mod n). The number base r is congruent to (n + 1), conversely, the digit n | (r − 1). Alternatively, n | ω, or is an omega-related totative of base r. This sub-case will be colored light blue. The other sub-case r ≡ −1 (mod n). The number base r is congruent to (n − 1), conversely, the digit n | (r + 1). Alternatively, n | α, or is an alpha-related totative of base r. This sub-case will be colored light green. Thus, Case 2 is a special case of Case 1, a residue ρ coprime to n with n a neighbor-related totative of base r. Throughout the entire span of residues ρ (mod n), there is only one alpha-relationship, ρ = (n − 1), and one omega-relationship, ρ = (n + 1). All other coprime residues ρ or cases gcd(ρ, n) = 1 will have digit n an opaque totative in corresponding bases r.

Case 3: 1 < gcd(ρ, n) ≤ n. If the residue ρ is not coprime to n, then n will at least be semi-coprime to r. This case will be colored yellow.

Case 4: Let the integer β be the product of the distinct prime divisors of r, and let the integer ν be the product of the distinct prime divisors of n. The digit n is regular to base r if and only if β = ν, additionally, n does not divide r. We might say the residue ρ is regular to n if and only if β = ν. We can consider Case 4 a special case of Case 3, since having any common prime divisors is possible only when 1 < gcd(ρ, n) ≤ n. This case will be colored orange

Case 5:  r ≡ 0 (mod n). This case has the number base r congruent to zero (mod n). This means r is some integral multiple of n, kn. Thus, n | r (i.e., n is a divisor of r), and gcd(r, n) = n. Note that throughout the entire span of residues ρ for the digit n, this is the only instance of n dividing r evenly. This case will be colored red.

Conjectures based on observations.

1. The value n = 0 actually signifies that an arbitrary integer a is congruent with r, i.e., a ≡ 0 (mod r) and doesn’t represent a “normal” digit. Instead, n = 0 represents cases where r ≡ 0 (mod n)

2. If n = 1, n both divides and is relatively prime to r. The single residue for n = 1 will be colored purple.

3. If n is prime, then all residues ρ are coprime to n except ρ = 0.

Figure 1: n = 11 (Prime)

0 1 2 3 4 5 6 7 8 9 a

4. Let the integer e > 1 be an exponent of a prime p. If n is a perfect power of a prime divisor pe (cf. OEIS A001597), then residues that are multiples of p are regular to n.

Figure 2: n = 9 (Prime Power)

0 1 2 3 4 5 6 7 8

Figure 3: n = 16 (Prime Power)

0 1 2 3 4 5 6 7 8 9 a b c d e f

5. If n is squarefree composite (cf. OEIS A120944), then (nonzero) residues ρ that are not coprime to n can only have n semi-coprime to r.

Figure 4: n = 10 (Squarefree)

0 1 2 3 4 5 6 7 8 9

Figure 5: n = 30 (Squarefree)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

6. If n is the product of at least one prime power pe, with e > 1, (cf. OEIS A126706) then those residues ρ that share all elements of the same set of distinct prime divisors with n will be regular to n.

Figure 5: n = 12

0 1 2 3 4 5 6 7 8 9 a b

Figure 5: n = 18

0 1 2 3 4 5 6 7 8 9 a b c d e f g h

Looking at the digit maps digit-wise, we see a heat map of commonality of a residue ρ (thereby a number base rρ (mod n)) with the digit n. When the residue ρ = 0, r is congruent to n and n | r. The residues ρ coprime to n are gray and those that are not are at least yellow. If a residue ρ shares with n all the elements of the set of distinct prime divisors, it is regular to n and is colored orange. The residue ρ = (n + 1) is omega blue and ρ = (n − 1) alpha green.

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icarus
Posted: Sep 14 2012, 09:57 PM


Dozens Demigod


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The Abbreviated Multiplication Table

Abbreviated Multiplication Table for r = 60
showing patterns of relationships between products and the base

01 02 03 04 05 06 07 08 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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 00
03 06 09 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 00                    
04 08 12 16 20 24 28 32 36 40 44 48 52 56 00                              
05 10 15 20 25 30 35 40 45 50 55 00                                    
06 12 18 24 30 36 42 48 54 00                                        
07 14 21 28 35 42 49 56                                            

Let the integer r ≥ 2 be a number base. Consider an unabridged multiplication table Mr of base r to be one that displays a 2 dimensional matrix of all the pairwise products of digits 0 < nr. Such a table has (r² + r)/2 unique products. Generally, we can abbreviate Mr such that this minimized table can be used to aid the Complementary Divisor Method of multiplication and division. Let the integers d, d′ divide r evenly. The CDM (formerly known as the Reciprocal Divisor Method, described here) relies on the formula

r = d · d′

Examples: r = 10 = 2 · 5 = 1 · 10, r = 12 = 3 · 4 = 2 · 6 = 1 · 12, etc. CDM works best in mid-scale highly composite numbers, where there are at least several pairs of complementary divisors d, d′. We can multiply by d by multiplying by r/d′ : in base twelve we can find the product 3(8) by using 10(8)/4 = 80/4 = 20 (dozenal figures). This is a less efficient process than “direct multiplication”, i.e., multiplication referring to a memorized Mr. In number bases much larger than r = 16, this becomes too cumbersome for most people. Using the CDM, we can multiply sexagesimal 3(40;24) by using 1,00(40;24)/20 = 40,24/20 = 2,01;12. We can reach nondivisor factors by breaking either the multiplicand or multiplier into a sum of more convenient parts. Sexagesimal 41(43) can instead be broken into 2(20(43)) + 43, for instance. This practice is called an offset.

In general, an AMT can be produced 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 60, with 104 unique products, 189% the size of the full decimal table and 133% the size of the 12× table commonly learned in grade school. The sexagesimal AMT is nearly ideal, with a solid range of small divisors d < r½, supporting multiplicative offsets up to 6×. The AMT is generally well-penetrated by the dense set of divisors, filling the products involving the 6 smallest positive integers with a solid block of regular products, relegating nondivisor totatives to 8 isolated unique products, 7 of these in the 1× table (i.e., are prime). The sole composite totative product in the table is digit-49. If we discount the index (the 1× products), we have a table of 74 unique products, falling between the 55 pure decimal and 78 12× table products. For this reason, I believe decimal-coded sexagesimal might be considered an irregular human-scale number base (base 6×10). The long-standing Babylonian usage of sexagesimal tends to support this idea.

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icarus
Posted: Oct 5 2012, 03:17 PM


Dozens Demigod


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Regular Product and Coprime Product Tables.

We can abbreviate the multiplication table of number bases in the transitional range at the upper side of the human scale (i.e., composite bases roughly between 16 and 30) in a couple different ways. The bases in this range may have full multiplication tables that prove a bit large for general human arithmetic. The abbreviations can be used in conjunction with the relatively small AMT to supplement multiplication in a pure base in this range. Where the AMT is a truncation of the full multiplication table of base r, these abbreviations are a sort of lossy compression of the full table. We will use octodecimal (base 18) to exemplify these abbreviations.

Patterns in the Full Octodecimal Multiplication Table

1 2 3 4 5 6 7 8 9 a b c d e f g h 10
2 4 6 8 a c e g 10 12 14 16 18 1a 1c 1e 1g 20
3 6 9 c f 10 13 16 19 1c 1f 20 23 26 29 2c 2f 30
4 8 c g 12 16 1a 1e 20 24 28 2c 2g 32 36 3a 3e 40
5 a f 12 17 1c 1h 24 29 2e 31 36 3b 3g 43 48 4d 50
6 c 10 16 1c 20 26 2c 30 36 3c 40 46 4c 50 56 5c 60
7 e 13 1a 1h 26 2d 32 39 3g 45 4c 51 58 5f 64 6b 70
8 g 16 1e 24 2c 32 3a 40 48 4g 56 5e 64 6c 72 7a 80
9 10 19 20 29 30 39 40 49 50 59 60 69 70 79 80 89 90
a 12 1c 24 2e 36 3g 48 50 5a 62 6c 74 7e 86 8g 98 a0
b 14 1f 28 31 3c 45 4g 59 62 6d 76 7h 8a 93 9e a7 b0
c 16 20 2c 36 40 4c 56 60 6c 76 80 8c 96 a0 ac b6 c0
d 18 23 2g 3b 46 51 5e 69 74 7h 8c 97 a2 af ba c5 d0
e 1a 26 32 3g 4c 58 64 70 7e 8a 96 a2 ag bc c8 d4 e0
f 1c 29 36 43 50 5f 6c 79 86 93 a0 af bc c9 d6 e3 f0
g 1e 2c 3a 48 56 64 72 80 8g 9e ac ba c8 d6 e4 f2 g0
h 1g 2f 3e 4d 5c 6b 7a 89 98 a7 b6 c5 d4 e3 f2 g1 h0
10 20 30 40 50 60 70 80 90 a0 b0 c0 d0 e0 f0 g0 h0 100

Abbreviated Multiplication Table for r = 18
showing patterns of relationships between products and the base

1 2 3 4 5 6 7 8 9
2 4 6 8 a c e g 10
3 6 9 c f 10      
4 8 c g          

Regular and Coprime Product Tables for r = 18

Regular Products Coprime Products
1 2 3 4 6 8 9 c g 10
2 4 6 8 c g 10 16 1e 20
3 6 9 c 10 16 19 20 2c 30
4 8 c g 16 1e 20 2c 3a 40
6 c 10 16 20 2c 30 40 56 60
8 g 16 1e 2c 3a 40 56 72 80
9 10 19 20 30 40 49 60 80 90
c 16 20 2c 40 56 60 80 ac c0
g 1e 2c 3a 56 72 80 ac e4 g0
10 20 30 40 60 80 90 c0 g0 100
1 5 7 b d h
5 17 1h 31 3b 4d
7 1h 2d 45 51 6b
b 31 45 6d 7h a7
d 3b 51 7h 97 c5
h 4d 6b a7 c5 g1

Base 18 is small enough that an octodecimal culture may approach multiplication using two or three abbreviated tables, basically avoiding dealing with semicoprime products for the most part. We can abbreviate the octodecimal multiplication table two different ways. The first is a table of regular products g that are the result of multiplying two regular numbers g, these regular products and numbers have distinct prime divisors that are limited to those of the base r, one or both of {2, 3}. Such products are rather common in daily life; knowing the regular products brings the mnemonic size of the table down to 55 unique products, exactly the size of the pure decimal table learned in grade school. Folks might then “offset” multiplication problems by shifting multipliers and multiplicands to regular numbers. Suppose we want to multiply octodecimal “d” (digit-13) by 8. We can rewrite (“d” × 8) as (“c” × 8) + 8. Then we would remember from the regular product table (RPT) that (“c” × 8) = “56” (decimal 96). Adding 8 to “56” yields the answer: “5e” (decimal 102). Thanks to the three octodecimal divisors that nearly fill the range less than the square root of 18, we can reach any point on the full multiplication table of base 18 using offsets of k, 2k, or 3k, since the widest range of octodecimal digits not covered in the octodecimal RPT is 3 digits wide (i.e., the range between digit 13 and digit 15 inclusive).

If society saw it fit to reduce the occasion of offsets, some might memorize a small table of the products of the 6 octodecimal totatives. This coprime product table (CPT) has 21 unique products. If one memorized this table, one could offset from the products of any combination of octodecimal regular digits and totatives to reach products of semitotatives. Suppose we want to multiply octodecimal “e” (digit-14) by 5. There are two methods, we will use the method involving totatives to illustrate. We can rewrite (“e” × 5) as (“g” × 5) − (2 × 5) = (“g” × 5) − “a”. From having memorized the CPT, we would know (“g” × 5) = (“h” × 5) − 5 = “4d” − 5 = “48” (decimal 80) and subtracting “a” from this result would yield “3g” (decimal 70) as the answer. We could also have avoided totatives and arrived at the same conclusion. We can rewrite (“e” × 5) as (“g” × 5) − (2 × 5) = (“g” × 5) − “a” just as before. From having memorized the RPT, we would know (“g” × 5) = (“g” × 4) + “g” = “48” (decimal 80). Subtracting “a” from this result would yield “3g” (decimal 70) as the answer.

The quantity of coprime products may be so few that it doesn’t merit memorization of an octodecimal CPT as it might in tetravigesimal.

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icarus
Posted: Oct 5 2012, 03:22 PM


Dozens Demigod


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Coded Bases (i.e., Mixed Radices)

Let the integers {r, s} ≥ 2, with s | r (i.e., base s is a divisor of base r). A coded base r is one which uses a smaller “sub-base” s to break down the size of an r that has arithmetic tables out of mnemonic reach by the average person. Even when we are not intending to perform arithmetic in a large base r, we may be using such as an auxiliary base β in a cultural base r, as we use decimal-coded sexagesimal in our decimal world. Generally we can regard the coding of a base r by s as beneficial when we preserve the distinct prime divisors p of base r in base s. However, this shouldn’t serve as an absolute criterion. We know there is a grouping base s′ such that

r = s · s′,

and we might also find three complementary divisors that together produce r. We will only consider coded bases that have only a grouping base s′and a sub-base s, so that in effect we represent each digit of base r as a two-figure expression. The units are represented by a lower unit-figure or unit-staff in base s, and the groupings of s are represented in the upper figure or upper staff in base s′. In our civilization we use decimal-coded sexagesimal with decade-figures in senary (base 6) and unit-figures in decimal.

When we have a perfect square base r that has a sub-base s in the human scale range of bases, we will usually see a devolution to that base s when we intend to use such a base for human arithmetic. Thus bases {49, 64, 81, 100, 121, 144, 169, 196, 225, 256} will tend toward the use of bases {7, 8, 9, 10, 11, 12, 13, 14, 15, 16}. This is because the representation of each digit of the perfect square r as two figures in the same base s is really just using base s, with its simpler arithmetic tables and rules.

Decimal-Coded Digits of Base 18, using 3-on-6 notation, primes in boldface type

Wide Coded Base (18) Square Coded Base (36) Tall Coded Base (120)
  _0 _1 _2 _3 _4 _5
0_ 0 1 2 3 4 5
1_ 6 7 8 9 10 11
2_ 12 13 14 15 16 17
  _0 _1 _2 _3 _4 _5
0_ 0 1 2 3 4 5
1_ 6 7 8 9 10 11
2_ 12 13 14 15 16 17
3_ 18 19 20 21 22 23
4_ 24 25 26 27 28 29
5_ 30 31 32 33 34 35
  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9
0_ 00 01 02 03 04 05 06 07 08 09
1_ 10 11 12 13 14 15 16 17 18 19
2_ 20 21 22 23 24 25 26 27 28 29
3_ 30 31 32 33 34 35 36 37 38 39
4_ 40 41 42 43 44 45 46 47 48 49
5_ 50 51 52 53 54 55 56 57 58 59
6_ 60 61 62 63 64 65 66 67 68 69
7_ 70 71 72 73 74 75 76 77 78 79
8_ 80 81 82 83 84 85 86 87 88 89
9_ 90 91 92 93 94 95 96 97 98 99
a_ a0 a1 a2 a3 a4 a5 a6 a7 a8 a9
b_ b0 b1 b2 b3 b4 b5 b6 b7 b8 b9

A good sub-base s tends to lie close to r½, since the bases are comparable in scale, but this as well is not necessarily a hard-and-fast rule. Typically we would like a composite sub-base s. Usually the grouping base s′< s; when this is so, we will call such a coded base a “wide” or “horizontal” coded base. Because we want s < 16 (roughly speaking, so that it has arithmetic properties that people can use) we may find that a “tall” or “vertical” arrangement is convenient. When we have tall coded bases, we will see digits in the upper figure (that is of a base s′> s) which do not appear in the unit figure. An example is decimal-coded base 120; the decade figures are duodecimal and the unit figure is decimal. Two additional digits (digits ten and eleven) are required in the decade figures that never see use in the decimal unit figures. The digit range of the upper figure will serve as a weak but significant constraint on the size of the grouping base s′. This implies that the largest practical coded bases r may be around r = 240.

Octodecimal example:

Decimal-Coded Digits of Base 18, using 3-on-6 notation, primes in boldface type

Digit Map Divisibility Tests Divisibility Tests with SPD
  _0 _1 _2 _3 _4 _5
0_ 0 1 2 3 4 5
1_ 6 7 8 9 10 11
2_ 12 13 14 15 16 17
  _0 _1 _2 _3 _4 _5
0_ 0 1 2 3 4 5
1_ 6 7 8 9 10 11
2_ 12 13 14 15 16 17
  _0 _1 _2 _3 _4 _5
0_ 0 1 2 3 4 5
1_ 6 7 8 9 10 11
2_ 12 13 14 15 16 17

Usage of all three abbreviated multiplication tables would aid arithmetic in pure octodecimal, but the employment of senary-coded octodecimal would perhaps be the most efficient way for people to perform arithmetic in base 18. Unlike in other bases, the coding base 6 has the same distinct prime divisors as the base, so there is no “blunting” of natural divisibility by missing prime divisors as we see in decimal-coded sexagesimal or quinary-coded vigesimal. The downside of senary-coded octodecimal is that it halves the octodecimal concision, i.e., from about 79.67% decimal to about 159.3% decimal, so roughly 3 coded figures will be written in the place of every pair of decimal digits.

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icarus
Posted: Oct 5 2012, 03:58 PM


Dozens Demigod


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Auxiliary Bases

An auxiliary base β is one that aids the base-in-play r in resolving (i.e., rendering as an integer) the commonest fractions of a cyclical or unit measure. Common applications for auxiliary bases β are measures of angle or time (especially the day), or of proportion similar to decimal percent. The strategy for a good auxiliary base is to maintain a desired multiplicity ε for each prime divisor p native to the base-in-play r, while resolving key coprime primes q that are commonly-encountered fractions. A fraction 1/n (with the denominator n either p or the target q) resolved as an integer in base r that has a single significant digit is termed “clean” or “cleanly resolved”, and there is a range of “cleanliness” associated with the resolution of some fractions with more than 1 significant digit. An example of a cleanly resolved fraction in decimal-coded sexagesimal is the third of an hour as 20 minutes, with the quarter hour as 15 minutes less “clean”, but just as clean as native decimal ¼ = 0.25. The model for a good auxiliary base is sexagesimal “cleanliness”, that is, cleanly resolved fractions with denominators {2, 3, 4, 5, 6, 10, 12, 15, 30, 60}, with less clean eighths and sixteenths. This model presumes we are interested in resolving the fractions that have empirically been desired by the de facto implementation of sexagesimal and related auxiliary bases in our decimal culture.

Decimal Auxiliary Bases

r = 10 β
n d

r 12 24 60 120 360
1 2 5 6 12 30 60 180
1 3 3 4 8 20 40 120
2   6 8 16 40 80 240
1 4 25 3 6 15 30 90
3   75 9 18 45 90 270
1 5 2 2.4 4.8 12 24 72
2   4 4.8 9.6 24 48 144
3   6 7.2 14.4 36 72 216
4   8 9.6 19.2 48 96 288
1 6 16 2 4 10 20 60
5   83 10 20 50 100 300
1 8 125 1.5 3 7.5 15 45
3  

375

4.5 9 22.5 45 135
5   625 7.5 15 37.5 75 225
7   875 10.5 21 52.5 105 315
1 16 0625 0.75 1.5 3.75 7.5 22.5
1 10 1 1.2 2.4 6 12 36
1 12 083 1 2 5 10 30
1 9 1 1.3 2.6 6.6 13.3 40

The decimal auxiliary bases {12, 24, 60, 120, 360} each supply a clean third while sacrificing some resolution of fifths. The dozen and the 24-hour day both resolve the third but render fifths a terminating single-place fraction. We can observe that a fifth of a day is not a commonly-referred interval, our culture largely steps around the fifth of a day. The exception is when decimal parts of a day are used in simple algorithms that do not accommodate hours. The 60-part hour and minute renders clean thirds and fifths, with the fifth accepting a little blunting; we need two significant digits to represent the fifth of an hour. The 60-part division requires eighths to accept fractional sub-units. If we use 120-part divisions, we also resolve a clean eighth. The decimal use of 360 as a measure of angle is well-chosen. We can resolve fractions with denominators {2, 3, 4, 6, 9, 12, etc.} in a single significant place, {5, 8, 10, 20, etc.} in two significant places, and {16, etc.} with terminating fractional parts. The bisection of the right and equilateral angles are 45° and 30°, respectively, which are rather tidy decimal numbers.

Duodecimal Auxiliary Bases

r = 12 β
      60 120 720
n d

r [“50”] [“a0”] “500”
1 2 6 26 50 260
1 3 4 18 34 180
2   8 34 68 340
1 4 3 13 26 130
3   9 39 76 390
1 5 2497 10 20 100
2   4972 20 40 200
3   7249 30 60 300
4   9724 40 80 400
1 6 2 a 18 a0
5   a 42 84 420
1 8 16 7.6 13 76
3   46 1a.6 39 1a6
5   76 31.6 63 316
7   a6 44.6 89 446
1 14 09 3.9 7.5 39
1 a 12497 6 10 60
1 10 1 5 a 50
1 9 14 6.8 11.4 68

Duodecimal intrinsically accommodates {2, 3}, so the objective of a duodecimal auxiliary base might be to extend resolution to fifths. The necessity of an auxiliary base in duodecimal might be very much reduced, and we might live with purely duodecimal divisions, stepping around the fifths and tenths. Attempting to use the decimal-friendly superior highly composite auxiliary bases {60, 120} in duodecimal results in sacrificing some resolution of thirds, which is second-nature in base 12, to obtain cleanly-resolved fifths. Such a sacrifice of a smaller prime 3 to obtain 5 seems too great a price, unless some desire to resolve fifths proves essential. The duodecimal number “500” (decimal 720) supplies clean natural fractions, fifths and tenths. Perhaps the duodecimal “500”-system would be applied to geometry, especially where we might expect pentagonal symmetry.

Tetradecimal Auxiliary Bases

r = 14   β
      12 60 420 840
n d

r “c” “44” “220” “440”
1 2 5 6 22 110 220
1 3 49 4 16 a0 160
2   94 8 2c 160 2c0
1 4 37 3 11 77 110
3   a7 9 33 187 330
1 5 2b 2.58 c 60 c0
2   58 4.b2 1a c0 1a0
3   85 7.2b 28 140 280
4   b2 9.85 34 1a0 360
1 6 249 2 a 50 a0
5   b94 a 38 1b0 380
1 8 1a7 1.7 7.7 3a.7 77
3  

537

4.7 18.7 b3.7 187
5   8a7 7.7 29.7 14a.7 297
7   c37 a.7 3a.7 1c3.7 3a7
1 12 0c37 0.a7 3.a7 1c.37 3a.7
1 a 158 1.2b 6 30 60
1 c 1249 1 5 27 50
1 9 17ac63 1.49 6.94 34.94 69.49

Tetradecimal (base 14), like decimal, features a gap between its prime divisors {2, 7}. Our objective for an auxiliary base in this case would be to deliver resolution of {3, 5} so that we would have a resolved range of {2, 3, 5, 7}. Like in decimal, we could sacrifice some resolution of sevenths to buy resolution of thirds and fifths. In base 14, we have a neighbor in 15 that can deliver both desired factors. We see that the decimal-friendly auxiliaries {12, 60} aren’t satisfactory. We can turn to tetradecimal “110” (decimal 210) perhaps to support thirds and fifths. The quarter is left with less resolution than thirds and fifths, and we still have slightly messy eighths. Perhaps this would be a good option for tetradecimal sub-units of time. Tetradecimal “220” (decimal 420) might be applied to geometry, as it resolves the natural fractions rather well. The tetradecimal auxiliary bases have a bonus over their decimal and duodecimal counterparts in that they resolve sevenths.

If we have bases that incorporate larger primes p > 7, we might accept more profound “blunting” to buy resolution of smaller primes. This might indicate that our number base is out of tune!

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Double sharp
Posted: Oct 23 2017, 03:33 PM


Dozens Disciple


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Joined: 19-September 15



QUOTE (icarus @ Oct 5 2012 @ 03:58 PM)
If we have bases that incorporate larger primes p > 7, we might accept more profound blunting to buy resolution of smaller primes. This might indicate that our number base is out of tune!

This in fact suggests that because primes above 7 become a natural part of factorisation at a scale too fine for any auxiliary use, bases with prime factors larger than 7 essentially act like these primes don't really exist for the purpose of auxiliaries, and simply have to use unmodified superior highly composites whether or not they gel well with the base (and they don't). (Which is a bit different from decimal, which can use unmodified superior highly composites, but gels well with them.)
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