Cyclic Primes (2)

Wells makes the interesting observation that we partition 142857 into its two halves i.e. 142 and 857 and then divide the 2^nd partition (857) by the 1^st (142) we get the quotient 6, with a remainder of 5.

So the number 6 is 1 less than the cyclic prime involved i.e. 7, while the remainder  5 is 2 less.

This is a property which universally holds.

For example for the next cyclic prime (17) the 1^st and 2^nd partitions are

05882352 and 94117647. And dividing 94117647 by 05882352 we obtain the quotient 16 and the remainder 15!

More complicated - though in their own way equally exciting patterns - can be seen through breaking the repeating full cycle of digits into more than 2 equal partitions.

One intriguing case arises when we divide into 4 equal partitions. This requires the repeating sequence of digits of the cyclic prime to be divisible by 4!

Up to 100 the only candidates that qualify are the cyclic primes 17, 29, 61 and 97.

So starting with 17 we now divide its 16 repeating digits into the 4 equal partitions

0588   2352   9411  7647

Now beginning with the last we divide 7647, 9411 and 2352 respectively by 0588 finding in each case the quotient and remainder.

We then get in the 3 cases respectively

13 and 3

16 and 3

4  and  0

Now these 6 numbers can be represented in general terms by the following grid

a₁₁   a₁₂

a₂₁   a₂₂

a₃₁   a₃₂

The following set of relationships now hold (which universally apply in all such cases):

a₂₁ is always one less than the cyclic prime in question. Therefore in this case of the cyclic prime 17,

a₂₁ is 16.

So,

a₂₁  = 16 (i.e. 1 less than the cyclic prime)

a₁₁ + a₃₁  =  17 (i.e. the cyclic prime)

a₂₁ –  a₂₂   = a₁₁ (i.e. 13)

a₃₁ = a₂₂  + 1 (i.e. 4)

a₁₁ +  a₁₂  = a₂₁  – a₃₂ (i.e. 16)

a₃₁  – a₃₂  = a₁₂ + 1 (i.e. 4)

a₁₂ +  a₃₂  = a₂₂  (i.e. 4)

I then calculated the corresponding grid of numbers with reference to the cyclic primes 29, 61 and 97 respectively.  

These are

  p = 29              p = 61             p = 97

12     7              11     9             22    17

28    16             60    49            96    74

17     9              50    40            75    57    

Then as we can see the value for a₂₁ is always one less than the cyclic prime in question. It then becomes like a fascinating variation on a Suduko puzzle to fill in the remaining values on the grid using the general relationships I have listed (which always hold).

A number of possible incorrect variations however are consistent with the relationships given. This suggests that the values are determined in a simultaneous holistic manner!

However there is another interesting observation in that the ratios of terms in each row seem to approximate closely with each other.  So the closest relationships possible as between the ratios will correspond therefore to the correct entries in the grid.

The next cyclic prime with recurring digits divisible by 4 is 109.

So it would be interesting bearing in mind what I have said to predict the correct grid entries without manually calculating the values.          

A  simpler situation relates to three partitions which requires that the repeating decimal sequence of the cyclic prime be divisible by 3.

The first case therefore arises with the cyclic prime 7.

Again we break the repeating digits into 3 successive partitions, i.e.

14    28  57

Then starting with the last we divide 57 and 28 by 14 respectively, listing again the quotient and remainder in each case.

This time we get 4 and 1 in the first case and 2 and 0 in the second

So listing in a grid with general form

a₁₁   a₁₂

a₂₁   a₂₂

we get,

4              1

2              0

Then for the next applicable cyclic prime 19, the corresponding grid is

7             4

11           6  

Then the cyclic prime 61 the grid is

47              10

   13       2

What is clear in each case is that the sum of quotients i.e.

a₁₁ +  a₂₁  = p – 1

Likewise a₁₁ +  a₁₂  + a₂₂  = p  – 2.

Cyclic Primes

I mentioned cyclic primes in another blog contribution recently.

Many of their ingenious properties are to be found in that wonderful book by David Wells “The Penguin Dictionary of Curious and Interesting Numbers”.

So I will concentrate in this entry on some other interesting properties (not addressed in that book).

Clearly we operate customarily in the denary system (base 10). So full cyclic primes i.e. where reciprocals have a digital decimal period one less than the number in question) are expressed with respect to base 10.

So when we say that 7 is the first (full) cyclic prime, this is with respect to the base 10 system, where its reciprocal (1/7)  has a repeating decimal sequence of six digits (i.e. 124857). 

However the earliest base number for which 7 for example is also a (full) cyclic prime is in base 3 with the recurring digits 010212 and then again in base 5 (032412).

Then 5, which is not a cyclic prime in base 10 is however already fully cyclic in bases 2 (0011), 3 (0121), 7 (1254) and 8 (1463).

So if a prime number is not fully cyclic in base 10, it will be cyclic in a number of other bases.

In this sense a remarkable cyclic nature is shared by all prime numbers!

Incidentally 2, which is the first prime number is (fully cyclic) in base 3 (with just one repeating digit (i.e. 1). It is then fully cyclic in all odd number bases.

A simple formula can be given for the sum of digits of any cyclic prime.

If p is the prime number and n the base, then the sum of digits = (p – 1)(n – 1)/2.

So again 7 is fully cyclic in base 10. Therefore the sum of digits of its recurring period sequence = (6 * 9)/2 = 27.

Indeed this is related to another property.

Because the period (of all prime numbers other than 2) is necessarily even, we can partition the digits into two equal halves.

So with 142857 we have 142 and 857. When we add we get 999.

And this is a universal occurrence (i.e. where each digit of the sum of partitions is one less than the number base).

Now when we subdivide into smaller groups (which must be a factor) of the original partition size, the relationship likewise holds (except that we multiply by the other factor involved).

So if we partition in this case into single digit groups, this represents 1/3 of the original partition size. So the factors here are 1 and 3. Therefore when we sum the terms we get 3 * 9 = 27.   

This can be better demonstrated with respect to the recurring digit sequence of the next (fully) cyclic prime in base 10 i.e. 17.

So this has a period of 16 with the recurring digits 0588235294117647.

Now when we partition digits into two equal groups of 8 and add we get 99999999.

However we could equally partition into equal groups of 4 and add to get 2 * 9999. (The factors here are 4 and 2!)

We could the partition into equal groups of 2 and add to get 4 * 99. (The factors are now 2 and 4).

Finally we could partition into groups of 1 digit to get 8 * 9 = 72 which is necessarily the sum of digits of this cyclic prime! (The factors here are now 1 and 8). 

When one subtracts the original partition values an interesting result ensues.

So 94117647 – 05882352 = 88235295.

The first 7 digits here are a recurring sequence within the original digital sequence with the last digit 5 (representing the sum of the next 2 digits in the  full sequence (i.e. 4 + 1).

And this is a universal feature of such behaviour.

For example the 18 digits sequence for the next cyclic prime (19) is

052631578947368421.

Then, partitioning into two halves of 9 and subtracting we get,

947368421 – 052631578 = 894736843.

So once again the sub-sequence of 9 now contains 8 digits of the original sequence with the last digit (3) representing the sum of the next two digits (2 + 1) in the original sequence!

142857 * 142857 = 20,408,122,449

Wells points to the interesting fact that if we place 0 in front of the first digit we can then partition into two equal groups of 6, which when added obtains the original number 142857 (i.e. a Kaprekar number).

When we multiply 142857 by its reverse (758241) we get 108,320,034,037.

What is perhaps more surprising is that when we partition into two equal groups of 6 and add, we once more get 142857.

I say surprising, because the reverse number here 758241 does not in fact maintain the same cyclic sequence of digits!

However once again this is a property that universally holds for all cyclic primes across all bases (though not necessarily with original cyclic sequence of digits). .

Another interesting property of cyclic primes is a great regularity in the composition of digits. All the digits 0 – 9 are included before the pattern repeats.

For cyclic primes with period of (10n + 2) where n = 0, 1, 2, 3,  …., all 10 digits will appear at least n times with the two extra digits 3 and 6 occurring n + 1 times.

23 for example is a cyclic prime. This means that all 10 digits (0 – 9) will occur at least twice in the full period of 22 digits with 3 and 6 occurring 3 times.

Now cyclic primes in base 10 cannot have a period of 10n + 4 digits as this would imply that the prime ends in 5!

For cyclic primes, with period of 10n + 6 digits all 10 digits will again occur at n times with the 1, 2, 4, 5, 7 and 8 occurring n + 1 times.

Again for example 17 as we have seen is a cyclic prime. This means that 0, 3, 6 and 9 occur once with 1, 2, 4, 5, 7 and 8 occurring twice.

For cyclic primes with period of 10n + 8 digits all digits but 0 and 9 will occur n + 1 times as verified in the case of 29 (which is a cyclic prime).

This means that it is possible to immediately predict the number of times each digit occurs for any cyclic prime!

For example 47 is a (full) cyclic prime.

Therefore we know that its unique period of 46 digits contains 0, 3, 6 and 9 four times, and all the other digits i.e. 1, 2, 4, 5, 7 and 8, five times.

This of course implies that the sum of digits = (0 + 3 + 6 + 9) * 4 + (1 + 2 + 4 + 5 + 7 + 8) * 5 = 72 + 135 = 207 = (p – 1)(n – 1)/2 = (46 * 9)/2.

Here is another interesting fact related to octagonal numbers that I discovered some years ago!

The prime number 3 is cyclic in all bases 3n + 2 where n = 0, 1, 2, 3, ….

For example when n = 2 this means that 3 is cyclic in base 8, with the two digits i.e. 25 continually repeating it its decimal sequence.

Now the remarkable property of the number 25 is that when we subtract from its reverse (i.e. 52) we once again obtain 25 (in base 8).

And this property holds in all the (given) bases for which 3 is cyclic! 

In base 2, the reciprocal of 3 (as cyclic prime) has the recurring digital sequence 01

Then in base 5, it is 13, in base 8 (as we have seen) 25, in base 11 it is 37 and so on.

So for example, to move from the cyclic prime in base 5 to base 8, we multiply the first digit by 2 then subtracting 1 from the second digit, multiply by 2 and add 1.

Then in base 11 we multiply the first digit (with respect to base 5 by 3, again subtracting 1 from second digit and this time multiplying by 3 before adding 1.

Now when we convert these numbers 01, 13, 25, 37,….to base 10, we get

1, 8, 21, 40, …., which are the octagonal numbers!  

Tapestry of the Number System

I mentioned yesterday, in another of my blogs (Pop Memories) my appreciation of Don McLean’s songs (especially with respect to the quality of the lyrics that these songs contain).

In particular I singled out his song “Tapestry” in this regard. This provides both a wonderfully poetic description of the holistic interdependence of creation and the grave threat that is now posed to our environment through economic mismanagement of the planet’s resources.

Now you might think that this has nothing whatsoever to do with Mathematics, but certainly from my perspective you would be fundamentally wrong in this regard.

The current mindset that dominates economic thinking (with its emphasis on the exploitation of natural resources) is firmly rooted in our limited view of science (that attempts to view the environment in a detached objective manner).

And this view of science in turn is rooted in an ultimately distorted appreciation of Mathematics that misleadingly attempts to divorce quantitative from qualitative reality.

And this likewise leads to the attempted separation of the arts and the sciences (which again from an overall holistic perspective is most unhealthy).

Though I would maintain that the arts and sciences represent separate domains corresponding to distinctive methods of understanding, they do so in a merely relative sense. So ultimately from an overall human perspective they represent complementary pursuits (that are necessary for each other's integrated development).

Thus when we adopt an experiential perspective that is truly holistic, we no longer attempt to compartmentalise disciplines in a rigid manner.

Therefore in this spirit, I am suggesting to you that obtaining the fundamental insight to appreciate what the Riemann Hypothesis is truly about, can perhaps more easily come directly from artistic rather than scientific - or indeed direct mathematical - appreciation.

Indeed in this regard I suggest that you listen with a fresh mind to a song like “Tapestry”.

In connection with the Riemann Hypothesis, mathematicians often use the analogy of the music of the primes. However because of their compartmentalised training, they never seem to make the connection that music represents essentially a qualitative - rather than quantitative - mode of appreciation.

Put another way, it represent a holistic appreciation of the qualitative interdependence of various constituent notes, chords, instruments etc. rather than the strict analytic interpretation of each individual item in a quantitative manner.

However as it stands, Conventional Mathematics is totally lacking a qualitative dimension (in formal terms). Alternatively it is totally lacking a genuine holistic means of appreciating its relationships.

And as the mystery of the Riemann Hypothesis essentially relates to the marvellous  interdependent aspect of the number system, Conventional Mathematics remains simply incapable of recognising its true nature.

So it is like in a detective story where all the evidence is available to solve the crime, but no one can make the obvious inferences.

Now, though admittedly written in an artistic – rather than scientific – context, the opening lines of “Tapestry” are very revealing.

“every thread of creation is held in position

By still other strands of things living."

So in the context of our holistic appreciation of the number system, the threads here refer to the prime numbers.

Now these prime numbers are held in position by the (non-trivial) zeta zeros. And I have been continually making the point that there are in fact two complementary sets of zeta zeros (Zeta 1 and Zeta 2).

So these zeta zeros play a true holistic function in perfectly shadowing the prime numbers (which - relatively - are of an independent analytic nature).

In other words they ensure that the primes are locally distributed in just one optimally possible manner, so that their (relatively) independent nature at this local level, is yet fully compatible with their overall holistic interdependence in collective terms.

So in this way we have individual unique notes (the primes) comprising a cosmic number symphony (with not one note out of place). And this is because the precise positioning of the primes is holistically determined by the zeta zeros.

We could equally maintain from the reverse perspective that there is a holistic nature to the primes (as a collective whole) which ensures that not one of the non-trivial zeros (as relatively separate) is out of place!

As I have continually emphasised in my blogs, the holistic nature of the primes comes from their dimensional nature (which indirectly is expressed in a circular manner).

So, for example, we can view the prime number 3 in two ways:

1) in a linear cardinal manner. Here 3 =  3¹ (where the quantitative number 3 is raised with respect to 1 as representing its default qualitative dimension).

2) in a circular ordinal manner. Here 3 = 1³  (where the default quantitative value 1 is raised to 3 as representing its varying qualitative dimension). Indirectly this qualitative dimensional notion of 3 is expressed in a circular quantitative fashion through the 3 roots of 1).

So the primes (which must always be considered in two-way relationship to the natural numbers) keep dynamically switching as between cardinal (quantitative) and ordinal (qualitative) meanings.

Now the lyrics of Tapestry refers to “other strands of things living”.

This leads to the greatest surprise of all regarding the number system i.e. that from an authentic dynamic perspective, it is indeed living!

The reason for this remarkable property is that properly understood, numbers do not exist in some abstract world, but rather as the fundamental inherent encoding of all living phenomena.

So through incorporating the qualitative with the quantitative aspect of number, one’s whole appreciation of the nature of number radically switches to viewing it as the intrinsic encoded nature of all living systems. And from this perspective, everything in nature - from the most minute sub-atomic particle up - is understood as necessarily comprising part of a living process!

So perhaps we can now appreciate a little better the tapestry of the number system (as a dynamic living system with twin analytic and holistic manifestations which can switch depending on context).

This of course also implies that the number system is an intelligent system.

Again the very reason why we find this so difficult - even abhorrent - to appreciate is because we have become so used to looking at number in a distorted (merely quantitative) fashion.

So again the mystery of how the primes can be so perfectly distributed throughout the number system, points directly to its holistic intelligence (manifested through the zeta zeros) which ensures such synchronicity. Thus in a very important sense, the fundamental nature of the holistic intelligence that pervades all creation is thereby enshrined in the zeta zeros. 

I will quote part of another verse from “Tapestry” which may be instructive.

“Where the birth and the death of unseen generations
are interdependent in vast orchestration
and painted in colors of tapestry thread.
When the dying are born and the living are dead."

Now again in the context of the number system, the unseen generations here can be taken to refer to the unlimited nature of the zeta zeros.

Though many billions of these have now been detected (and inevitably will keep increasing), this represents but a very small set of all zeros.

So the interdependence of the primes with the number system is in fact dependent on all these unseen generations of zeta zeros which act to ensure perfect synchronicity.

So again this is how we can locate the “tapestry threads” i.e. the individual primes  (through the background holistic activity of the zeta zeros).

Now the last line is very instructive here:

“when the dying are born and the living are dead”

This essentially points to the ultimate paradoxical nature of the relationship of the primes to the natural numbers.

The very appreciation of interdependence implies paradox (from a dualistic perspective). If the primes and natural numbers are truly interdependent - which indeed ultimately they are - this implies that the primes in a sense are contained in the natural numbers and the natural numbers in turn contained in the primes. So from the cardinal perspective each natural number represents a combination of prime factors; however equally in reverse from the ordinal perspective each prime represents a combination of natural number members! 

So in this sense the primes (ultimately) are seen as perfect mirrors of the natural numbers and the natural numbers as perfect mirrors of the primes. 

And the only way of truly seeing this relationship is through avoiding any rigid identification in understanding with either perspective (in isolation).

So in this highly dynamic appreciation, the primes, as mirrors of the natural numbers, only can live briefly in experience through letting the alternative complementary perspective (of the natural numbers as mirrors of the primes) momentarily die.

Then the corresponding dynamic appreciation of the natural numbers as mirrors of the primes can in turn only live briefly through again letting the alternative perspective (of the primes as mirrors of the natural numbers) momentarily die.

Thus ultimately the pure experience of this mutually interdependent relationship is ineffable!

I will finish up by quoting the last verse of “Tapestry”

“Every fish that swims silent, every bird that flies freely,
every doe that steps softly.
Every crisp leaf that falls, all the flowers that grow
on this colourful tapestry, somehow they know.
That if man is allowed to destroy all we need.
He will soon have to pay with his life, for his greed.”

This clearly is pointing to the holistic nature of intelligence that pervades, in a fully natural manner, all plant and animal life on the planet.

The remarkable assertion here is that human beings uniquely possess the capacity to behave in an unnatural manner, thereby having the potential to destroy all that is in nature.

Now the intellectual roots of this unnatural behaviour lie deep in the nature of our distorted mathematical appreciation. And it is such appreciation that directly informs in turn our scientific and economic thinking!

And as we have seen, though again most readers again might find this abhorrent to accept, the ultimate reason for our capacity for such unnatural behaviour lies in this deeply confused interpretation of the nature of the number system (and by extension all mathematical understanding) where we have attempted - wrongly - to completely divorce qualitative from quantitative type appreciation.

So we have lost any concept of the global holistic nature of intelligence (as the collective unconscious of all life forms).

And again this is because we have attempted to portray Mathematics and the sciences - misleadingly - as representing merely conscious activity.

We are thereby reaching a true crisis now on our planet where due to this great imbalance, we are in increasing danger of being  swamped through ever more frequent economic and environmental disasters.

Much as we might hate to admit it, the key intellectual cause of all this problem lies in our extremely limited interpretation of the nature of the number system (and by extension all mathematical relationships).

Many continue to look on Mathematics as the one area of truth that can remain safely insulated from the contamination of global change.

In fact, properly seen, Mathematics (or rather our false understanding of Mathematics) is the root intellectual cause of this growing contamination!