**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,

** page 70, [F] pages 86-87.**