Origin of the Primes (1)

Origins of the primes

Since all natural numbers are primes or products of primes, except 1, it must be the primes that hold the key to the structure and the interaction of numbers. (Prime numbers are only divisible by themselves and 1)

If we hold that numbers are produced by preceding numbers and that all numbers are essentially the symbol of a ‘sum’ of ´ones´, the ‘one’ they are all divisible by, like 2 is 1 + 1 and 2 : 1= 2, likewise 3 is 1 + 1 + 1 etc. , but if we also hold that numbers, if not primes, are a unique product of primes, then it is easier to appreciate that plus or minus 1 makes all the difference in a non-primal number, even if it is a big number, because the ‘character’ of the number changes dramatically, not only changes the number from odd to even or vice versa, but, if not changing to prime, also the factors of the number change completely.

Factorisation defines every number’s unique character and factors are always primes, although here powers of 10 will be seen as an exception (as a root-prime of sorts) and can be expressed in factors 10^n ( ten to the power n; n is any number)

The difference in ‘factors’ between 1000 and 1001 is telling : 10 x 10 x 10 and 7 x 11 x 13

That 10 is not a prime but the product of 2 and 5 makes it a special number, because 2 and 5 are the only primes that have no period in the reciprocal (= inverse, like 7 and 1/7).

The period is a repetitive returning pattern of numbers in the inverse of the prime (1/7 = 0.142857 142857 …..), but with 2 it is ½ = 0.5 and with 5 it is 1/5 = 0.2 and no repetition, because they are each others reciprocal, so only in their case the inverse ends immediately and results in the other as reciprocal number, only in their case the product of prime and reciprocal period number is not 10^n – 1 (99….99, nine-only numbers, rep-digit), but 10 , no more, no less.

Because of this outstanding feature these two primes are called here ‘extra-primes’ (extra-ordinary) as opposed to ‘proper-primes’, that is, all other primes, except for 3, which is also an ‘extra-prime’, but here also called the ‘root-prime’ of all ‘proper primes’, it is also the root of 9 , the pivotal number of this whole calculus.

This difference in character of the first three primes and the rest is so fundamental, as we shall see, that most general rules that are formed here hold only for the ‘proper primes’ and that 2 , 3 and 5 are not seen as of the same category, although 3 , 9 and 27 shape all the ‘proper primes’, and 3, as a root-prime, produces itself as ‘proper prime’ through division on 10 (0.33333….)

3 is the one and only prime number that is identical with its inverse period-number and that shares most rules with the ‘proper’ primes, the order of which it creates and shapes.

The inverse of 3, its reciprocal, is in decimal expression 0,3333333……, but the period length is only 1 figure (0.3 3 3 3 3 3) and the produced period-number is 3, the ‘ proper root prime’. So 3 is extra-prime, root-prime and proper-prime. ( 3 x 0.33333….= 0.99999999…. = 10)

The period-number multiplied by the prime produces 3 x 3 = 9, the first nine-only number, and this 9 is 1 short of 10 and so the division repeats, the repetition, the period, the ‘number-wave’, is created, and infinite.

By squaring the root-prime 3 we get 9, then dividing the ‘square of 10’, a 100, by the square of 3 , the 9, produces another inverse , with a ‘hidden period’ because the period of the reciprocal is not obvious as it is in most cases, because it looks in decimals like 0,11111111………, so for 10/9 the period is 1. 1 1 1 1 1 , but for 100/9 it is 11, 11 11 11 11.
The period-length here is found to be two figures, two ‘ones’ and the produced second ‘proper- prime’ is 11, the true starting number of the ‘proper primes’ and the first proper repunit R2. (R1= 1 is not really repetitive).

Eleven is the first number beyond the initial 10 digits, the first power of 10 + 1 and the first repetition of digit, the first repeated unit and called a ‘repunit’, a number consisting of only 1’s.
In symbol “R(n) “ or also 11…….11 , so 11 is the prototype repunit, and moreover the first repunit that is also a prime number, which are relatively rare ( R19 and R 23 are also, but then the next are R317 and R1031 and only a few more are known, but there must be ‘infinitely’ many; which feels ‘awkward’ though)

The cube of the root-prime 3 is 27 and dividing the cube of 10 , a 1000, by the cube of 3, produces the reciprocal period of 37,037037037…., the period-length is 3 figures, the period-number is 37 and here the third ‘proper prime’ is created, which together with the root-prime 3 form the next repunit after 11 (R2), which is 3 x 37 = 111 (R3)

The first three primes 2 , 3 and 5 are outside the body of general rules, as said, but the peculiar relationship of the three first primes produces and shapes the order of all other primes.

All proper primes are the result of operations of 3 respective powers of 3, the root-prime, on 3 respective powers of 10, the product of the extra-primes 2 and 5.

These three operations in three dimensions by three powers of three on three powers of ten set the stage for the order of the primes, where 9 is omnipresent as representative of 3 and is factor to all periods of proper primes taken as whole numbers, except the period of 3.

Rep-units

Being the symbol of the 9-number system that is integral to the primes and their order, the repunits, the number 9 is complement in the ratio 10 : 9, the ratio that dominates this calculus and the order of the primes (10/9 = 1.11111111………..)

Nine in a sense has two periods, as 9 on 10 it has the period 1, as 3^2 (9) on 10^2 (100) it has the period 11, this dual nature of the period of 9 provides it with a versatility that is capable of combining straight and squared categories in whole numbers for circles and squares alike

The number series 9-10-11 is crucial ( as is its square 81-100-121), and the same numbers in their bilateral combinations dominate the geometric structure of the primes, and that of squares and circles as well, the relation of 9 to 10 and 9 to 11 , as well as 10 to 11 is fundamental in this logic, and in the logic of the primes

The rudimentary and very first relations are

10 : 3 = 3 + 1 (remainder)

10^2 : 3^2 = 11 + 1

10^3 : 3^3 = 37 + 1

10^9 : 3^4 = 12345679 + 1 = (37 x 333667) + 1

What we see here is the formation of the basic proper primes 11 and 37 , they, as factors of R2 and R3 are connected in deep-structure to most primes, via the repunits, like the numbers 2 , 3 and 5 are connected as factors to most natural numbers (over 70%)

But there is more here, there is a structural evidence for a fourth geometrical dimension, because the powers of 3 can easily be related to the powers of 10, only in the case of the fourth dimension , the powers of ten here are the cube of the third dimension values of powers of ten (10^3)^3 = 10^9 , but we also see a whole number correspondence between 9^2 (81) and a power of 10:
81 x 12345679 = 999 999 999 + 1 = 10^9

A nine-only number (999…999, rep-digit) divided by 9 always produces a one-only number (111…111) , a rep-unit (or repunit = repetitive-unit), so these nine 9’s correspond to the repunit 111 111 111 (R9), and is here its ‘ordinal’ number.

Many repunits are composites of smaller repunites and new primes as we shall see

The divisors of the first three powers of 10 are the first three powers of 3, the ‘root-prime’-10 is divided by root-prime 3 and produces ‘proper-3’ and a remainder of 1; 10/3 = 3 , rem. 1

100 is divided by 9 and produces the proper prime and repunit 11; 100/9 = 11 , rem. 1

1000 divided by 27 produces the (extra-ordinary) proper prime 37; 1000/27 = 37 , rem. 1

We know that a remainder of 1 in division is the sure sign of a period in the reciprocal, it is the 1 where the division starts all over again ad infinitum

It is easy to see why prime divisor and period must always have a product of 10^n – 1, a number of only-nines, like :

9 x 11 = 99  ;     27 x 37 = 999    ;    101 x 99 = 9999 ,

but also 7 x 142857 = 999 999 (R6) and 239 x 41841 = 9999 999 (R7),

this is because divisor and result reach a product of 99……….99 and not of 10^n, 100….000, and so create the remainder of 1 for the next division which creates the period, the cycle, the wave.

The length of the period of the prime’s reciprocal can never exceed in digits the number of the prime minus 1 (this is a mathematical necessity) and the period must, to be a period of a prime, secure a remainder of 1 in a division on a power of 10 , and must, multiplied by the prime, always produce a number of 9’s only, 999………999 (= 10^n –1)

Since the prime cannot be divisible by 9 as a matter of definition and the product of prime and period, the nine-only number, is always divisible by 9, it must be the period of the proper prime that is always divisible by 9, because no new factors get added.

So the rule is that the period of the proper prime (’s reciprocal) is always divisible by 9 and never by 2 and 5 (the factors of the powers of 10) and that, once the period-number is divided by 9, it becomes a deep-structure number. This deep-structure number is a product of primes which, when multiplied by the prime produces inevitably a repunit.
The primes are tied together in this base-repunit and will not occur other than in this combination in other repunits.
Combined they are a set of factors characterizing their base-repunit and defining their relations.

Every proper prime is factor to a repunit, inevitably. Q.E.D.

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This piece is continued in ‘Repunits and waves’ where the whole thing gets translated into geometry and waves.

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