I wrote this paper from 1994-1996, for the fun of it. As to whether or not it has to do with anything in real life, don't ask me. -William Ager
Properties of Prime Sets that Repeat after N Digits in their Inverse
I. Introduction
When the inverse of any rational real integer is taken, there is finite real number that it repeats. This repeated number, as with all numbers, can be broken down into its constituent parts, or prime numbers, which can not be broken down farther. It is clear that there is a basic relationship between a number and the number it repeats in its inverse:
In equation 1.1, R is the repeated number, "places" is the number of digits R has, and "number" is the number that produces R in its inverse. So 1/number=.RRRRR...
Although Equation 1.1 is very general, it is not fundamental. This is because any number (like number) can be broken down into its primes. Therefor we need an equation that shows the relationship between a prime and the number it repeats. Beginning with Equation 2.1 we will no longer be concerned with what number repeats what, but rather the question to be answer is: Why a certain prime or set of primes repeat N times, and what are the properties of that set?
Here N is the number of digits in the repeated number. k is the number of primes in the set that repeats N times: PN, pk is an element of that set: pk Î PN, and s is the repeated number of the product of all of the primes in PN, with all of the factors of 2 and 5 divided out. s is also a number that is composed of the primes that correspond to the factors of N, multiplied by 9 (this is the most practical way to obtain s ). Here, all the real rational numbers are dispensed with, and their components are used instead. This is because the number that is repeated by any number just repeats the same number of times as the prime with the highest corresponding N. Individual primes within the set PN also have this exact relationship:
Where pk¢ is a particular prime in the set PN, and rk¢ is its repeater with all factors of 2 and 5 divided out. Even though all the primes within PN have this property, it will be useful to define a special prime that can be distinguished from the rest. After putting Equations 1.1 and 1.2 together, we get:
Equation 2.3
Since all of the primes we choose to be particular have this property, the whole set can will have the same property:
Summation: 
Equation 2.4
Multiplication: 
Equation 2.5
Equations 1.4 and 1.5 say nothing new, other than the fact that the sum and product of a particular prime and its repeater is the same for every other sum and product within the set. From Equation 1.3 and 1.5, we can obtain an Equation that can pick-out a particular prime:
k>1 
Equation 2.6
For special cases of k values Equations 1.1-1.6 take on more simpler forms: Where p1<p2<p3, and r1>r2>r3.
|
k=1 |
k=2 |
k=3 |
|
|
|
|
We can state several general conclusions that come from the equations above, and what I have said so far. The only exception is Postulate 4 which has not been proven:
|
· Postulate 1: |
s = the product of repeating primes that correspond to the factors of N multiplied by 9. Exceptions include: N=1, N=22 |
|
· Postulate 2: |
If N is prime, then s =9 and Õ p=1.111eN. Conversely, if N has many factor, then s is large and Õ p is small. Hence: s µ 1/Õ p |
|
· Postulate 3: |
If k=1, then s =r=Õ r |
|
· Postulate 4: |
N multiplied by an integer plus one is always equal to pk: pk=akN+1 (where ak is any integer 1 or greater) |
TABLE OF THE FIRST 21 ORDERED N PRIME SETS
|
N |
Õ pk |
p k |
r k |
s |
a k |
k |
|
1 |
3 |
3 |
3 |
3 |
2 |
1 |
|
2 |
11 |
11 |
9 |
9 |
5 |
1 |
|
3 |
37 |
37 |
27 |
27 |
12 |
1 |
|
4 |
101 |
101 |
99 |
99 |
25 |
1 |
|
5 |
11111 |
41 271 |
2439 369 |
9 |
8 54 |
2 |
|
6 |
91 |
7 13 |
142857 76923 |
10989 |
1 2 |
2 |
|
7 |
1111111 |
239 4649 |
41841 2151 |
9 |
34 664 |
2 |
|
8 |
10001 |
73 137 |
1369863 729927 |
9999 |
9 17 |
2 |
|
9 |
333667 |
333667 |
2997 |
2997 |
37074 |
1 |
|
10 |
9091 |
9091 |
1099989 |
1099989 |
909 |
1 |
|
11 |
1.111111e10 |
21649 513239 |
4619151 194841 |
9 |
1968 46658 |
2 |
|
12 |
9901 |
9901 |
100999899 |
100999899 |
825 |
1 |
|
13 |
1.111111e12 |
53 79 265371653 |
188679245283 126582278481 37683 |
9 |
4 6 20413204 |
3 |
|
14 |
909091 |
909091 |
109999989 |
109999989 |
64935 |
1 |
|
15 |
90090991 |
31 2906161 |
32258064516129 344096559 |
11099889 |
2 193744 |
2 |
|
16 |
99999967 |
17 5882353 |
5882352941187583 1700000561 |
100000033 |
1 367647 |
2 |
|
17 |
1.111111e16 |
2071723 5363222357 |
48269001213 18645507 |
9 |
121866 315483668 |
2 |
|
18 |
999001 |
19 52579 |
52631578947368421 19018999981 |
1000999998999 |
1 2921 |
2 |
|
19 |
1.111111e18 |
1.11111e18 |
9 |
9 |
NA |
1 |
|
20 |
99009901 |
3541 27961 |
28240609997175939 3576409999642359 |
1009999999899 |
177 1398 |
2 |
|
21 |
900900990991 |
43 1933 10838689 |
23255813953488372093 517330574236937403 92262080773791 |
1109999889 |
2 92 516128 |
3 |
