Explaining primes


The question of randomness of the prime numbers, numbers only divisible by themselves and one (2, 3, 5, 7, 11, 13 etc.), I believe to have solved 6 years ago with strictly logical reasoning concerning the reciprocals of every prime number, that is, their inverse value. (like 7 and 1/7= 0.142857 142857 14…….)
This solution I have published now as ‘Repunits, order of the primes’, being a 6 year old text I once sent to Nature and others.

The essence is the insight that the reciprocals of the primes lead us to the frame-work of their order. It are these reciprocals and their character that define this order of the primes, a system they lay bare even in several layers.
There may be other, more polished, ways to show the order in the primes, but this solution is all-embracing, transparent and flawless.
It also points to the deeply intricate relation between 10 and 9, as this shows itself in a mechanism that creates primes in a certain logical order ( as I show in older publications elsewhere). But we have to see through these two numbers, 10 and 9, the primes 2 and 5 = 10 and the square of 3 = 9 at work, creating a new, second, dimension.

It is the interaction of these first 3 primes that creates and frames the order of the rest, I have called them ‘extra-primes’, because they stand outside all the others which I have called the ‘proper’ primes. These terms I use in older work and within this frame work there is nothing wrong with these names.
Together with the original text of my solution to the order of the primes, I will publish here this more elaborate piece which is easier to understand because it works with concrete examples and because I have a better grasp of the logic now.

The key to the order of the primes is in their reciprocals.

The crucial numbers in my solution are the prime reciprocals which turn into the so-called ‘Repunits’. A rep-unit (repetitive unit) is a number which consists of only ‘ones’ and the symbol is R with an ordinal n, so Rn, is the general symbol of the repunits and R1 = 1, whereas R2 = 11 and R3 = 111, R4 = 1111, etc. So this is all simple enough, we can also write it as 11……..11, with any number of ones in between.
There is though another way of expressing this and it is {(10^n)-1} / 9 = Rn, so this means any power of 10 minus 1 divided by 9 gives a repunit, 10^3 = 1000 – 1 = 999 divided by 9 is 111 (R3), etc..

Another structural feature is that any repunit with an ordinal number which is an even number and bigger or equal to 4, can be written as a product of a repunit and a power of 10 plus 1, so 11 x 101= 1111, and R2n = Rn x {(10^n)+1} or 1111 = R4 = R2 x (10^2) +1; then: 1111 x 10001 = 1111 1111 (R8) ; etc.
We see in this immediately the predominance of the decimal system. Because of this the two primes 2 and 5 which form the product number 10 are the only primes which are not factors of repunits, but all other primes are, and are so necessarily, as will be shown.

In this model primes are created by the interaction of 3 , 2 and 5, that is: the product of 3 and 3 interacts with the product of 2 and 5, these 3 fundamental primes create all the ‘proper’ primes by the operation of 9 on the powers of 10, through the difference in number of only 1 ; the term ‘proper’ meaning that these primes share a common origin in the combinations of the first three ‘extra- primes’, as will be shown.
The crucial point is that we find all the primes as factors of either 10 ( only 2&5), or all powers of 10 plus or minus 1, and this frame work is hidden in the deep-structure of the repunits, which look quite featureless, but embody an intricate system in which the logic of the primes flows out in an ever expanding wave, as it seems.

To one side the wave is in an infinite series of floats, as ‘ones’ at sea level, to the other side it is an infinite line of zeros, at sea level, starting with a float and ending in an ever retreating float, again, of course, at sea level. (Riemann Hypothesis has zeros at sea level; I’ve no idea what it means, but this way I stay in character).
The powers of 10 plus 1 look like this: 1000000………..00000001, so an infinite amount of zeros and, as you see, definitely at sea level.

The repunits occur as derivation from the relation between 10 and 9. The crucial point is that only 1 less in a power of 10, reverts it into a 9-only number, a rep-digit, since 10000 – 1 = 9999 etc.
 So it is immediately clear that the 9-only number (99999….9999) can always be divided by 9 and will always produce a number of only 1’s. This is the repunit, here 9999 / 9 = 1111, or R4, which then in this case is the product of 11 and 101 , that is, the first and second power of 10 plus 1, 10+1 = 11 ; 10 x 10 + 1 = 101, both are primes.

I thought I ever had the proof why no other powers of 10 plus 1 can ever be primes and that proof may have to do with this system, maybe even with the fact that the repunits sometimes are primes, but I know I lost the answer quite quickly because I only later realized how important it was. The conjecture is that no power of ten plus one can ever be a prime, for n bigger than 2, because it must be necessarily composite.

Very few repunits are primes, but there are a few already in the beginning starting with R2 = 11, and then R19 and R23, also prime-repunits, but together not even 10 are known for sure today, I believe.
 (There have been found much more now, 2013) Nevertheless we see here that the prime-repunit is a structural aspect of the whole building of repunits. We will see that many logical aspects of the numbers involved in this repunit-logic occur as primes themselves in places where other primes cannot provide the necessary product. (9091, 909091 etc.)
It shows convincingly that the whole structure has definite layers in which the numbers relate, which is the proof of the order. (mathematicians probably see this as ‘accidental’)

How do we know that 10 itself together with all the powers of 10 plus or minus 1 hold all the primes as factors?
As follows: 
Except 2 and 5 which are factors of 10, all other primes are factors to repunits, because all primes produce 9-only numbers, by multiplying the period of the reciprocal by the prime, this period has seldom one, but usually several primes as factors. ( 7 and 1/7 = 0.142857 142857…, so 7 x 142857 = 999999 = R6) This is a unique combination of primes which produces exactly that product, which is a repunit, no matter what. These primes then do not occur in the system without their original co-primes, by which they are linked in the rep-unit or rep-zero. This is a remarkable quality.

There is though a structural quality in the repunits which makes that every repunit with an even ordinal number bigger than 2 is the product of a repunit and a power of 10 plus 1, I call a ‘rep-zero’, thus : 11 x 101= 1111 (R4); 111x 1001= 111111 (R6), so rep-unit times rep-zero = twice rep-unit. (rep-zero: 1000….0001)
This is structurally so and cannot be otherwise and thus it means that, since all repunits hold all the proper primes, part of these primes must belong to the product with numbers characterized by a power of 10 plus 1, which are the repunits with the even ordinals, so half of all rep-units holds a rep-zero and is of the form 111…111 x 1000…0001 = 111111……111111.

This is remarkable because it shows that the primes 2 and 5, combined in 10, split the world of the primes in two, that is, those primes that are factors to powers of 10 minus 1, the repunits, and those that are factor to all powers of 10 plus 1, 10000…..000001, the rep-zeros, so the difference of only 1 either side of powers of 10, structures all the proper primes without exception in two different classes.
This is the basic division between proper primes, they are always factor to a power of 10 , either plus or minus 1, so they are factor to either rep-units (11..11) or rep-zeros (100…001).

Repunit-factors are 11 (11); 3, 37 (111); 41, 271 (11111); power of (10^n +1)-factors or rep-zero-factors, are: 101 (101); 7, 11, 13 ( 1001); 73, 137 (10001).
These are nearly all very important and/or early primes, especially 37 and 137; how close they are, then 37 x 37 = 1369 + 1 = 1370 = 137 x 10; 41+1 = 42 = 6 x 7; 73 +1= 74 = 2 x 37 ; 137-1 =136 = 8 x 17 or 137 +1 = 138 = 23 x 6; 271+1 = 272 = 16 x 17

All primes except 2 and 5 have a period in their reciprocal, this is their inverse value. This period is an ever recurring number or sequence of numbers, which stretches endlessly. This is also the hallmark of a ‘rational number’, a number that expresses the ratio between two whole numbers, which the reciprocal is.
So 1/7 = 0.142857 142857 142857 …… etc. and this sequence 142857 is the period of the reciprocal of prime 7, but it also, as period, shows, multiplied by the original prime, a product of 7 x 142857 = 999999, that is 10 to the power 6 minus 1, this 9-only number divided by 9 gives the repunit R6 = 111111.
In fact the ordinal of the repunit signifies the amount of numbers in the period of the prime, which it usually shares with other numbers, unless it is a unique period prime

The period number 142857 is a proper number in its own right and can be factorized, but because it produced the 9-only number through multiplication by the prime, which is never divisible by 9, we know this means that the period of the reciprocal number is always divisible by 9, otherwise it could not produce a 9-only number with a prime, which it necessarily does.
142857 / 9 = 15873 and this number, which I call the ‘deep-structure-number’ of the prime, is the product of 3 x 11 x 13 x 37, so the primes 3, 7, 11, 13 and 37 are united in the Repunit R6= 111111, but this repunit can also be seen as the product of 111 x 1001, where the factors get split in 3 x 37 (111) and 7 x 11 x13 (1001)

We see now that already in R6 all early primes are factor and we find the perfect rule that from hereon all primes will be factor to the repunit with an ordinal number 1 less than the prime. So 7 is necessarily factor to R6 (6= 7-1), and so is the prime 97 factor to the repunit R96, etc…

We therefor also know that all repunits with ordinal under 96 together hold at least all the prime numbers under 96 as their factors, because all the earlier primes were factor in the repunit with ordinal number 1 less than the prime number.
This rule goes even further in that the ordinal number is often half the number of the prime minus 1, so the prime 31 is already factor in R15 ( 15 + 15 = 30 + 1 = 31).
This is even more telling when we see that the big primes in a repunit number relate to the ordinal number of the repunit in the following manner.
All the prime factors to a repunit, each plus or minus 1, will be multiples of the ordinal number of the repunit.
So the primes are not only fixed in the repunit- (or rep-zero-) product but also relate to the ordinal of the repunit by adding or subtracting only 1.

Here we see another layer of order when we examine the factors of the repunits and add or subtract 1 from the prime factors, this will always result in a number which is divisible by the ordinal number of the repunit. In this way all numbers are related to certain primes, which will always occur in the repunit with any number as ordinal number.

Say, you wonder which primes relate to the number 23, then you start multiplying 23 with, 2, 3, 4, 5 etc. and to this product you add or subtract 1, in our case 2 x 23 = 46 + 1 = 47, which is indeed a prime, but also 6 x 23 = 138 + 1 = 139 (but also -1 = 137), 110 x 23 = 2530 + 1 = 2531 and 23904225412692 x 23 +1 = 549797184491917, these 5 primes will always occur with each other as factors of repunits, never go alone in other combinations, they are linked by their deep-structure-numbers.
 This holds for all primes in a completely fixed framework, only those numbers and no other!

We see that the primes are completely embedded in a super-structure of logic which emanates from the powers of 10 plus or minus 1, so the number nine, ten and eleven as 10 minus 1= 9, 10 and 10 plus 1= 11 are the absolute basic numbers here, which are the very numbers my mathematical model, ‘the Celestial Proportion’, comprising of ‘the Rainbow Proportion’ and ‘the Pyramid Proportion’ are based on.

The analysis of the primes was the beginning of the number-logic which is the red thread of this website.


To find the lists of factors to subsequent repunits go to the Internet, there are several lists, but there are also lists which divide the rep-units and rep-zeros; this is one : http://gmplib.org/~tege/fac10p.txt , more you don’t need, really, but what you need to really appreciate this number-logic is a good calculator, like my blessed MapleV4, which has given me so much insight in numbers. I hope to organize some out prints of my Maple here on site.

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