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Kodegadulo 
Posted: Jan 8 2014, 02:34 AM


Obsessive poster Group: Moderators Posts: 4,184 Member No.: 606 Joined: 10September 11 


Kodegadulo 
Posted: Jan 8 2014, 02:40 AM


Obsessive poster Group: Moderators Posts: 4,184 Member No.: 606 Joined: 10September 11 


Kodegadulo 
Posted: Jan 8 2014, 02:43 AM


Obsessive poster Group: Moderators Posts: 4,184 Member No.: 606 Joined: 10September 11 


Kodegadulo 
Posted: Jan 8 2014, 02:50 AM


Obsessive poster Group: Moderators Posts: 4,184 Member No.: 606 Joined: 10September 11 


wendy.krieger 
Posted: Jan 8 2014, 03:05 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
Awayness was originally used for something entirely different. Specifically it set a relation between bases. For example, 18 and 24 are at fives with each other, and 18 and 48 at sevens.
The relation of awayness is perfectly recriprocal. The awayness of 18 in base 48 is equal to the awayness of 48 in base 18, both being 7. This means, that in either base, the other stands not just on a line seven removed from the base, but one that contains seventh powers. Some product of 18 and 48, not involving seventh powers, gives a seventh power. For example, b18 2'3000 is the seventh power of 6, and in b48, one has 18.0.0.0.0.0 as the seventh power of 24. This means, that if the index of a prime (ie the p1 divided by the period), is a multiple of 7 in one, it is too in the other. It covers prime powers too, but the period is p^{ n1}(p1). The relation of bases is used as a guide for factorising algebraic roots of \( b^n 1 \). Fermat's little theorm tells you that Ax consists of primes of the form xy+1, and usually at most one repeater. Knowing the relation between two numbers helps things along. In dozenal, the bases 10 and 80 are at threes. The prime 81 is then has an index that is a multiple of 3 , because for dividing A2, it is 40. So 81 has an index that is even (quadratic recriprocal), and a multiple of 3, because it has the same in a base 10 is at threes with. So the maximum period is 80/2/3=14. But since the factors of A1, A2, A4 and A8 are known, and do not hold 81, we note that 81 divides a14, and thus has a 14 place period in dozenal. Likewise, with 16, which is at fives with 20 and at sevens with 40, one sees that 21 is 5*5 and 41 is 7*7, that for 5 dividing its own period in 20, and 7 in 40, then 5 and 7 must divide their own periods in 16 too. 
Kodegadulo 
Posted: Jan 8 2014, 03:30 AM

Obsessive poster Group: Moderators Posts: 4,184 Member No.: 606 Joined: 10September 11 
Wendy, perhaps you could demonstrate that with a few examples. I'm sure there would be math fans interested.

wendy.krieger 
Posted: Jan 8 2014, 05:10 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
The rule of lines is to single out the close regulars. I shall use decimal here, but it works in all bases.
A single place gives the dividers of 10, to wit, 1, 2, 5. Each of thses are multiples of dimes, divide the dollar. As multiples of the dollar, divide the eagle, and so forth. We could write these as 10. That is, for being multiples of one place, are divisors of the next. One can enclose any number of places in this way. 1000 gives those that are multiples of mm but divide the metre. There are certain mantissa that are new here, not found in eg 00. The order of a regular is then the minimal number of 0 fall between the bars. These might be arranged by mantissa. A factorseries is then all those less than order n. 
wendy.krieger 
Posted: Oct 18 2015, 06:38 AM


Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
Let's look as base 18 as an example. The regulars for base 18 are (opposite powers of 3) 1 6 2 C 4 16 8 2C G 56 These are numbers of the form 18^x 6^y, and if the number itself is not a proper power (like C0 is 6^3), then the relation between them is at 3's with. So it means that if two numbers are at x's with each other, then (p1)/period are the same. For example, with 12 and 18, the relation is threes, and you get this. Note that 10*8 is also at threes with 10.
The thing to note here is that the period of one base divides p1 to give a multiple of three, the other one does too. For example, the period of 157 in B12 gives 156/3, = 52, the base 18 gives 156/156 = 1. Note that with 19, B12 gives 3, and B18 gives 9. 

Kodegadulo 
Posted: Oct 18 2015, 08:12 AM


Obsessive poster Group: Moderators Posts: 4,184 Member No.: 606 Joined: 10September 11 
What do you mean by, not a "proper" power? Proper power of what? Looks like a perfectly proper power of 6 to me. It's also a perfectly fine example of the formula 18^x 6^y you just cited, simply substitute x=0 and y=3. You've got people thoroughly confused already.
You know, in mathematical discourse, most people would introduce a new and unfamiliar term for a new kind of relation by defining the general case first, before using an example of it. The relation is terms of some x, so where does the p come from?
Totally lost. Which of these numbers are supposed to be the bases, which the primes, and which the periods? I see a bunch of prime numbers, but is 9 supposed to be a prime? It's composite. This table does not line up nicely at all when viewed on a cell phone, nor when looking at it in the Quote editor on a desktop. Is it really too much to ask you to format something like this with a <table> tag in a [ dohtml ] block, and put some actual labels on your columns and rows?
At first I thought B12 and B18 were supposed to be based numbers in base 18, then I realized it was just your shorthand for saying "base 12" and "base 18". Come on now, is it really such a chore to type out the word "base"? 

wendy.krieger 
Posted: Oct 18 2015, 11:08 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
"Proper" is a pretty ordinary mathematical term, which means not including the unit case. For example, 60 is a proper divisor of 120, but 120 isn't.
A number like 12^1, is a power, but not a proper power, since it is the first power. Something like 6^6/18^3 is a proper power, since it is equal to a power of a smaller number: 2^3. Note that eg 24*18, given as 6^5/18^2, is 432, is not a proper power, since no number raised to something greater than the first, gives 432. The relation of at x's with, means that a^m b^n = c^x, where the gcd(x,m)=gcd(x,n)=1. This means that gcd(i(m), x) divides gcd(i(n),x) for all primes, where i(n) is (p1)/period(p,n) and period(p,n) is the periodlength of p in base n. The table showing primes 6n+1, the primes are in the columns, and the bases are in the rows, so the top half shows bases 12 and 18, for which 12*18 = 6^3, and the second shows 10 and 80, where 10²*80 = 20³. Note the presence of 9 in this list. The sevenites are included in this process as well, so if two bases are at a relation of 'fives' (such as 18 and 24) or 'sevens' (as 18 and 48 are), and 5 or 7 is a sevenite in 18, it is in bases 24 and 48 as well. 
Kodegadulo 
Posted: Oct 18 2015, 01:37 PM


Obsessive poster Group: Moderators Posts: 4,184 Member No.: 606 Joined: 10September 11 
Don't be patronizing Wendy. Even if we understand what "proper" means in principle, long experience has shown that you tend to mangle terms and leave out critical bits of context. For instance, while 2^3 may be a proper power of 2 it is not a proper power of 8. If you meant to say that the numbers need to be proper powers of _some_ integer, you could have said that, or better yet you could have said "x must be a proper power of some integer, i.e. there must be some integers m,n both greater than 1 such that x=m^n". It's best to provide both the English description and the precise mathematical specification.
You see, that's more like it.
I already have noted it. Why is it among a collection of "primes", when it is composite? There also seems to be some misalignment, so I couldn't tell if you meant the 9 to be a base or not. Really, if you can't be bothered to do some formatting to make a clear table, don't be surprised if your observations get little currency. 

wendy.krieger 
Posted: Oct 19 2015, 12:48 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
The terminology I use for bases derives from tablesin the CRC Handbook of Mathematics, from photocopies of pages out of L E Dickson History of Mathematics, and perhaps Yates 'Repunits', along with Wells 'Dictionary of interesting and curious numbers'.
This is woefully inadequate coverage, so i invented edtra terms. Suppose that in base b, the period of a prime p is n. For ordonary 'class 1' primes, n divides p1. The discussion is not around the period but the number of coperiods, that is how many different loops there are. This is the reduced index. CRC mentions a primiive root, the smallest number g for which if g^n = 1, then p1 divides n. The primitive root needs to be ajusted occasionally for sevenites, that is, if p^2 divides g^p  g then one moves onto the next These instances i call sevenites in g, g^i passes through every number from 1 to p1. One can tabulate i for each prime 219 etc, these add to get value i, the number of loops is gcd(i, p1), and the period is (p1)/gcd(p1, i). One calls base b/a as b^n  a^n, and this offers "algebraic roots", one different for each N. You can write these in algebraic form, or in base form. Primes 'saying' x may divide Ax. if p divides Aj, then p divides pApj, which means that p is saying pj too, and the entry might be used to handle sevenites at that order. If p and q are at some relation c's, then there is some p^a q^b = r^c, where gcd(a,c)=gcd(b,c)=1. Two numbers at the relation of c's, the looplengths of two primes a, b, follow the pevious relation. Note the relation of being at c's is transitive, and that if one of the two is a proper power, then the relation holds only for the complement. For example, 8 and 18 are sixes with each other. But base18[ 8 is a proper cube, so the actual relation is at twos. Likewise 80 is a proper square, and the relation is a cube, and 800 is a simple number (dec 2592), the relation is sixes.] The proper title of the table i give is p saying 3, and includes 9, for which the euler totient is the same as 7, and like 7, has a primitive root. 