The Repunit and the Wave


Origin of the Primes 2

Nine and Eleven

The period of 11 is 1/11 = 0,09090909……… and this period-number 9 is shared by all repunits, however big, they have only one 9 at the end of the period of the repunit’s reciprocal, it is this one 9 that all periods of primes share and the 9 that incorporates the 9-number-system in the decimal system

This feature stresses again the link between repunit and 9, that is best symbolized in the rational number 100 / 9 = 11, 11 11 11 11 11 11 11…………..

Exceptional here is that in this primal beginning it is not other primes that produce new primes, as is the soon emerging rule, but a second and third (dimension) power of 3, and this is because there are no other primes to do it yet and this is the best illustration of why 3 and 9 dominate the deepstructure of the primes : because they produce it.

All even repunits are always linked to 9 , the odd ones when multiple of 3 are linked to 27 that produce the first proper primes 11 and 37 in this way and lay the basis of the order and its rules

10 / 3 = 3 rem.1 — 3 x 3 = 9 / 9 = 1 R1

100 / 9 = 11 rem.1 — 9 x 11 = 99 / 9 = 11 R2

1000 / 27 = 37 rem.1 — 27 x 37 = 999 / 9 = 111 R3

11 is factor to 11 (R2) and is one of the rare repunit-primes, because repunits are usually composite numbers and here 111 (R3) has two factors, factor 37,( the third centred hexagonal number after primes 7 and 19) and root-prime 3

R2 and R3 are factor to every second (even) and third repunit so the majority of repunits has 11 (half of them) as factor and 3 and/or 37 or 333667 and occur in all repunits of ordinal multple 3, that is a third, so the vast majority (67%) of repunits has at least one of these early proper primes as factors.

Now that we have this system of repunits and primes we can see that R4, which is 1111, must be divisible by R2 and this produces the next prime 1111 / 11 = 101. (10^2+1)

This prime 101 is special as first prime in the hundreds, like 11 is in the tens, and it forms a new and decisive step in the formation and further presence of the order of powers of 10^n +1.

11 is the first, 101 is the second, but 1001 is the third and then suddenly we can even derive part of the primes from another stock as well , because 1001 is factorised 7 x 11 x 13 and here we find primes together that only occur again when 111 x 1001 = 111111 R6.

In general these are the primes that are factor in even repunits, because a major new rule is now that any repunit Rn multiplied by a power of (10^n) +1 , will always produce a repunit twice the initial one: R2n, so :

R3 x (10^3)+1 = R6 and formula (Rn x (10^n) +1 = R 2n)

One by one, with every step, we find new rules to apply and new numbers to incorporate in the body of the primes, a body that has a rather different character from that of the production of composite natural numbers, because the amount of primes becomes less and less and the amount of composite numbers grows at their expense and to find the primes is rather more cumbersome, than just adding one to find the next number, but we are greatly helped by the iron-cast internal structure of the repunits which makes the ‘majority’ of primes within easy reach of calculation, depending on the capacity of a computer

Prime ordinals
A repunit like R5 and all other repunits with ‘prime ordinal’ numbers like R7, R11, R13, R19, are more difficult to factorize and they add only new and apparently unrelated primes to the already calculated ones, like R19 and R23, which are among the rare repunits which are primes themselves.

R5 produces the primes 41 and 271 as factors which are 40 + 1 and 270 +1 , and doubtless here there is again order because 41 follows 37 as prime and 40 +1 is strongly connected with 36 + 1 = 37 and we can see 10 and 9 in there, as we see in 270, so there is lots to be found in different layers.

Primes often are part of a new substructure themselves, like 37 relates to 333667, seen as 3…….7 and the same can be traced in the prime repunits R317 and R1031 , which are reminiscent of 37 and 101, the earliest proper primes (3….7, 10……1)

[ even as co-factor in 37 x 333667 = 12345679 !! (100/81) ]

It is the R6 that really combines and prduces the early primes like 7 and 13 (7×13=91) as new primes and those are then related to R2 and R3, so we have

11 x 111 (37×3) x 91 (7 x13) = 111 111 (R6) or

111 x 1001 = 111 111

Like 37 also 7 is a centred hexagonal number and 13 is a centred dodecahedral (12-)number (ball-packing), this means that there is a 1 unit , a centre, surrounded by multiples of 6 or 12, like 6+1=7, 12 +1=13, 18+1=19, 36 + 1=37 etc. and since all primes are of the form 6n +/- 1 and (probably) half of them are thus of the form 6n + 1 , a substantial amount of centred hexagonal and dodecahedral numbers are primes, and relate to packing (densely) (Kepler’s conjecture)

The pentagonal dodecahedron is the symbol of a twelve-sided centred compression of a sphere, the rhombic dodecahedron (crystal) emerges from perpendicular field contraction and fills all space between its units, something the pentagonal dodecahedron cannot

All of these early proper primes have geometric qualities and they shape the order by their specific qualities and here we see that the hierarchy of the primes is not numerical but geometrical, these primes define geometric ordering principles, and it is this that defines their prominence and frequent occurrence in analysis and their weight in the ‘hierarchy’

The ‘early primes’ are all short period primes, but dominate the complete logic of the primes

This we will expand on presently, here we continue the analysis of the deep-structure of the primes


It is the repunit that forms the deep-structure of the prime, in the repunit the prime finds its connection to other primes, the repunit can be considered a ring of connected numbers that will always appear in combination, and it is essential to see this in perspective with the waves and their patterns

This ring is symbolised by the repunit 111111..……111111

which reveals its ring-like quality easier in the graphic notation ..….11111111111111111…….

‘Ones’ can be added in front, at the end, in the middle, the number doesn’t budge and expresses the same, an infinite number of ones can be added and so an infinite number of primes, because every new repunit produces at least one new prime, but can go up to 7 , but there is a definite decline here, because the numbers grow by powers of ten, whereas on average 4 primes are added in that amount.

Usually a repunit produces 3 to 5 new primes, so with every new power of ten on average only 4 new primes are added, so fewer and fewer primes in a sea of composite numbers.

The repunits are ever expanding rings of numbers, that, with every new 1 added, add more than a power of ten new numbers under their number and, as there is no end to the wave, there is no end to the numbers to describe it, so this is the infinity of the wave, of the period, of the prime.

Example of deep-structure

To give an example of analysis of the deep-structure of primes we analyse the prime 7

We start with calculating its reciprocal (inverse)
Its inverse is 1 / 7 = 0,142857 142857 142857 14……….

The length of the period is 6 digits and a new rule is that the length of the period of a prime’s reciprocal is always a number that is identical with the ordinal number of the ‘repunit of first occurrence’ (rofo), the base-repunit of the prime, in this case R6.

The period number of 7’s deep-structure is 142857 and it is this number we divide by 9 to find the ‘deep-structure-number’, in this case 15873 (7 x 15873 = 111111 = R6)

The next step is that we analyse on divisibility by 3 by taking the sum of the digits and see if they are a multiple of 3 , the sum is 24, so 15873 / 3 = 5291 and because of the extra divisibility by 3 (after being divided by 9 already) we know the prime 37 must be at hand, because 3 (here actually 27) will never appear ‘alone’, will always be accompanied by 37 (R3 = 111) , or 333667 (R9 as 1001001), and indeed 5291/37 = 143 , which is 11 x 13. The subsequent step is that we divide by 11, because R2 is factor to all even repunits, thus half of all repunits.

Only after these steps have been taken do we know what kind of repunit we have, is it a composite repunit or one with prime as ordinal, in this case dividing by 11 gives 143 / 11 = 13 and that is a new prime, so R 6 is the base-repunit, repunit of first occurrence, ‘rofo’ of 7 and 13

We now have localised and brought into contact all the first 6 primes; 2 and 5 as the decimal system that ‘creates’ the primes by forcing repetition in the inverse (circularity), 3 as the universal (repetitive) operator (divisor) (root-prime), together called the ‘extra-primes’, and then 7, 11 and 13, all in very close connection, called ‘early proper primes’ with small periods and factor to early short repunits, R2-R9.

We have found that the period length of the prime is always the same as the ordinal of the repunit, so if any prime is at hand, by calculating its inverse and getting its period length we also know to which base-repunit the prime belongs, and the ordinal number of the repunit then tells us if it is even or odd, if it is a composite ordinal or a prime ordinal and in this way of any prime its deep-structure is ready at hand, and can be the start of analysis.

Another feature of the repunits is that all its new factors, that is, those factors (primes) not related to earlier repunits, are equal to or multiples of the ordinal number of the repunit plus 1

Example of analysis

R11 has only factors : 21649 and 513239
so (21649 – 1 )/11 = 1968 and (513239 – 1)/11 = 46658

Here we can discern another layer in the deep-structure because 1968 is factorised , 2, 3, 41 and 46658 is 2, 41, 569 , so both carry 41 as factor in a deeper layer still, but this is of no direct concern to us here, it only shows that the numbers have deep layers, that do not show readily, because only now it is obvious that R11 is linked to 41 which in its turn is linked to R5 as factor, so there is more there, again, but what interests us here in this rule is that we can always find the factors of a repunit however difficult it may seem

Suppose we have a large prime-ordinal-numbered repunit , then if we create all multiples of the ordinal number + 1 between them all the factors of the repunit will be present, so the repunit defines its own primes, and deep-structure, by its ordinal multiples + 1

The deep-structure of the prime is the base-repunit which defines the length of its period and the other primes it is directly linked to in the hierarchy of the periods (and the repunits consequently), so the primes are each others irreplaceable co-factors in the repunit, in the reciprocal, so we see that the repunits are the order of the primes.

Although it may seem different, these primes define each other exhaustively in the products they form together, their place is as solid as that of 7 between 6 and 8, because 7 as a prime is inevitably linked to 13, 11, 37 in the R6

Returning to the important repunit R6, important because of the ´weighty´ early primes it combines, it reminds us of a hexagonal order and indeed the primes are hexagonal-connected, 7 is, like 37, a centred hexagonal number, while 13 is a centred dodecahedral (12-)number, the first scale of a ball-packing, a spherical order

So we see here in this early ring of crucial primes 7 (6+1), 13 (12+1) and 37 (36 + 1) together in R6, the base of the hexagonal grid that is further highlighted by all primes, except 2 and 3 (2×3=6) being of the form 6n +/- 1

All early proper primes have definite geometrical qualities, because here can be added 11 that is the key to squaring the circle and in this system 11 forms a tandem with 7 to cover all circles and squares, spheres , tori and cubes in whole numbers

This all underscores the claim that the primes are directly related to geometry, the geometry that rises from the repunits and that casts the primes in a definite geometrical frame with great regularity and transparency

The repunits in this geometry symbolize the infinite circle that ever expands and that holds all the primes as its factors, like Riemann’s zero’s at sea level, or his primes in one line.

That the primes and their deep-structure are more than only regular in repunits is shown by the convincing products that show ever more layers in the structure of the primes, like the primes that form the powers of 10^n + 1

11 = 11

101 = 101

7 x 11 x 13 = 1001

73 x 137 = 10001

11 x 9091 = 100001

101 x 9901 = 1000001

11 x 909091 = 10000001

17 x 5882353 = 100000001

7 x 11 x 13 x 19 x 52579 = 1000000001

19 x 52579 = 999001

again mostly the same numbers recur

bigger repunits are of R4 1111 = 11 x 101

structure R10 1111111111 = 11111 x 100001

or R12 111111111111 = 1111 x 100010001

R14 11111111111111 = 1111111 x 10000001

So the hierarchy and structure of the primes is very much related to the powers of ten plus or minus 1

(10^n) +/- 1

Different combinations of this same pool of initial primes, gives rise to numerous symmetric number patterns that can be of great help with computation

Because of the extreme regularities in the deep-structure of the primes all answers to computation already exist in this model, only the search to connect the intermediate answers is the work of the computer, and all numbers are based on natural numbers, so this model works extremely fast, already now, without substantial input, its factorisation beats Maple easily

It connects square and circle without the barrier of Pi, and this is how sphere surface and corus surface can both be expressed in whole numbers and whole number relationships
This makes the decimal system and its notation into not just another convenient contingent choice, no, this system has intrinsic qualities that are peculiar in relation to the primes, including the logic of its notation and symbol, it is because of the central role of 9, 10 and 11.

The decimal system stands out for transparent deep-structure, its capability of creating waves of numbers and numbers of waves

Squaring (and mutually multiplying differing) repunits gives characteristic wave-like number series

R2 x R2 = 121

R3 x R3 = 12321

R4 x R4 = 1234321

R7 x R7 = 1234567654321

R9 x R9 = 12345678987654321

One can interpret this last sequence as a complete wave, in which rarity and density come to a maximum in the lowest and highest numbers, and it is R9 itself that holds the key to the wave structure of the repunits.

Only in R9 times R10 123456789987654321, there is still symmetry, although the chain of alternation seems broken by two 9’s , seemingly still connected, but this connection stops with the following square repunit

R10 x R10 = 1234567900987654321

We find the 8 missing in the first sequence and this will remain the characteristic of that number sequence, because: 123456789 + 1 = 123456790

Moreover we now see also a definite characteristic ending of the squared repunit from R9 onward, the perfect number sequence : 987654321

So the squared repunit has the general form :

12345679 0123……….4320 987654321

The first half of this number 12345679 0123…. is the same as that of the inverse of 81 (that is 9 squared)

1/81 = 0, 012345679 012345679 012345679 012345…………

The second half of this number, the ending : 987654321 , is not only a remarkable number in its own right, it turns out to be the key of the geometry of the torus and it is the number that is the digital of the rational number 800 / 81, again the inverse of 81.

9,87654320 987654320 987………. ( here 987654320 + 1 = 987654321 )

It is this rational number that is a substitute of pi^2 (pi-squared) and taking the square root of this number you get a number that is about the same as p (3.14159…), that is : 3,142696805…….. = 0,00110…. more than Pi, it is also written 2,2222…..V2.

Note that these are all core values of this website and its unusual geometry !

Here maybe for the first time a square root number becomes a valid substitute for Pi , and it is called : Qute (pron: “cute”) , by me, symbol Q, because it is a ‘cute’ number, as will be shown, and because the Q, as symbol, so perfectly expresses what happens in its geometry as will be explained.

Two rings of prime numbers create, when squared or multiplied, a torus, a square, a rectangle

This is where the new mathematics takes off.

Number waves

We have seen how the square of a repunit reveals a new deep-structure 123456790…. and ….987654321 and both are centred on 81, the square of 9 , because

100/81 = 1,23456790 123…… and 800/81 = 9,87654320 987654320 987……

Given that the repunit as single number represents a circle circumference, then the squared circle, the squared repunit, is the image of a torus

The formula for the torus is 4.pi^2.k.r , but in this special case k = r, so we get the formula
4 . pi^2 . r^2 , the “corus”, which is the square of 2. pi . r , the circle

This special case of the torus, as squared circle, where all radials go through one point in the middle, is here called “corus” , contraction of ‘core’ and ‘torus’ , but is in mathematics called : ‘horn torus’, and its surface is equal to a square, the square of its radial circle circumference.

It is this square’s perimeter that is equal to the great circle of the ‘corus’, but also to the great circle of the sphere that encompasses the ‘corus’

The relation between corus and encompassing sphere, which is the same as ‘squaring the circle by circumference’, turns out to be a central issue in the geometry of the whole numbers and in the here exposed geometry and dynamics of space.

The repetition of the period in the squares of the primes beyond R9 is the sure sign of an existing period and is therefore another example of a wave, because the change in the numbers spreads from the centre of the number symmetrically to both sides while the outer ends of the squares of the primes remain the same, the circle widens but remains closed, the centre pulses the numbers to both sides which shows graphically:

R 9 ^2 = 12345678987654321

R10 ^2 = 1234567900987654321

R11 ^2 = 123456790120987654321

R12 ^2 = 12345679012320987654321

R13 ^2 = 1234567901234320987654321

R14 ^2 = 123456790123454320987654321

R15 ^2 = 12345679012345654320987654321

R16 ^2 = 1234567901234567654320987654321

R17 ^2 = 123456790123456787654320987654321

R18 ^2 = 12345679012345678987654320987654321

R19 ^2 = 1234567901234567900987654320987654321

R20 ^2 = 123456790123456790120987654320987654321

R21 ^2 = 12345679012345679012320987654320987654321

If we do not square but multiply different order repunits, that is different number circles, then we see a different pattern because the ‘number wave’, is characterised by the smallest repunit of the two

R3 x R 6 = 12333321

R3 x R 16 = 123333333333333333333321

R8 x R 10 = 12345678887654321

R11 x R 12 = 123456790110987654321

R11 x R 13 = 1234567901110987654321

R11 x R 21 = 1234567901111111111111111110987654321

Because these last series is all to do with unequal circles we are dealing with a proper torus here, that seems to express another type of wave and its surface is a rectangle in 2 dimensional space.

Nine number order

Another feature that is fundamental is that the repunits follow a 9-number order, because the patterns of the numbers are the same for R9, R18, R27, R36 etc, so we see that the 9-number pattern that was already visible in the origin of the primes as factors of repunits, now as a second wave seems to propagates through the numbers, not through one number individually, but through the whole body of numbers

There is no better way of visualizing the wave than the nine-number system pulling through the decimal order by lagging behind one point on every ten, it is supposed that the intricacies of the two wave systems creates resonance and standing waves and that the ratio of 10 : 9 is the quintessence of the numerical and geometrical order that is shown here

It is the order of the natural numbers
It secures that resonance is discrete and sustains the standing wave that everything is made of.

We have become familiar now with the fact that it is possible in mathematics to find all kinds of numbers that not only express harmony in a hidden way, but even show it in and by themselves

With this knowledge, that the elegance of numbers expresses something of ‘real’ visual beauty and harmony, an objectivity that is even more striking in our perception of musical versus false tones, we may better appreciate why the geometry of the natural numbers that is presently revealed was born of aesthetics and found, and still finds, one of its major arguments in what it shows: harmony

Whatever this harmonious model may mean in practical use time will tell, but the fact that it is possible to construct such an ‘eternal’ model in the first place and then even apply it successfully to the solar system, as I show, should in itself be a fact of considerable importance to science.

That the geometry used here so strikingly resembles the geometry of the old Egyptians, the formidable builders of the Giza site and pyramids, who show to have been aware of the crucial ‘early’ primes, 11, 37, 101, 137, that only adds to the overall suggestion that something of great value may have been unearthed here again.


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