This is an indication of packing and asymptotic behavior. The fairly spreader 5 is in lead, followed by the most efficient packer 9.
As the number of primes nears the asymptotic line; distribution of all digits within a prime (i.e. M49 above) should equal among and for all primes (see Total of M1-M49 below). To this effect; primes distribution at the top of this page to some degree reverses. McCranie`s result on first digits 1,3,7 (see propulsors below) as most frequent (in this order) for the first 10 billion primes may support this.
The table below shows that all Mersenne Last Digits (besides for M1) are either 1 or 7. All Mersenne primes with First Digits of the boson-like (see Shake 2) triangular sub-group 3,6,9 have Last Digits 1 (and SLDs of the graviton-like pentagonal sub-group 1,5,7).
In exampel, the reduced number (digital root) of 11 is 1+1=2. Before proposing a new way of understanding prime behaviour of what is called the building blocks of numbers; let us consider the numerical evidence for Shake 7b.
The figure above shows frequency of reduced number 1-9 for 500.000 prime numbers.
These are divided into 10 sub-samples of 50.000 primes. The first sub-sample spans all first digits 1-9.
The other nine runs solely at one of the first digits 1-9. Column headings are marked with dark
grey to point out first digit 5 as the midth between 1 and 9.
The main point is that no primes have reduced number 3, 6 or 9 (since indivisible by 3).
Nor will any Mersenne prime (2^n − 1) have reduced number 3, 6 or 9 for n.
Other observations and hypotheses:
Reduced Number 5 - which is the midth of 1-9, obtains both global maximum and minimum for primes at first digit 1 and 5. Having both extramals, reduced number 5 obtains a middle position for both first digit 9 and in the Sum column for all Reduced Numbers. This indicate that 5 has a quality of fairly spreading. Reduced number 4 and 5 are "flatteners", the jumping champion 4 being the roughest with most maximums.
There are no maximum or minimums for primes running at first digit 6. Are 6 so "beautiful" that it never touch max or mins?
7 has a busy job as a maximum (horizontal) and its own global maximum (vertically) in the first column, where primes are running on all first digits 1-9. Are 7 an equaliser? Are 2 an 7 the most "receptive" root values (also as their high frequency in the Plimpton tablet can have been used for sieving out the necessary lengths to achieve angles with only trigonal reduced numbers)?
We observe that primes running at 3 and 7 as first digits - symmetrical placed around the "fairly spreader" 5, have the same frequencies. Further; primes running at first digits of the quadratic sub-group 2, 4 and 8 have their highest frequency for reduced number 8. 8 also tops the Sum column. Are 8 an "imposed boundary"?
Fantasy or numerosity? Maybe these syntetic algebraic steps can lead to an analytic understanding of the whole. In mathematics, numbers have their distinct de re qualities. The sub-groups (1,5,7), (2,4,8) and (3,6,9) have distinct qualities shared within the group. Which sub-group a higher number belongs to is found by the reduced number. By knowing the multiplication (windings of the circle group), you can do the reverse tracing.
So, on intuition - as the counting goes on numbers grab into each other in a more and more complex way. Sequence space theory should add to the analytical understanding. The complexity is fractal, but it has important denominators as Reduced Number, First Digit and for primes known Last Digits. How about seeing the flow of primes through pipes of discrete expanding pipe volumes (repetition of First Digit 1....10, .....100, ....1.000, ....), nearly causing turbulence at Reynolds number 3/7 = 0,42857142857142857... clearing out 3, 6 and 9? A flux combination of Benford`s Law and the fine structure constant.
In addition to the 500.000 primes in the figure above, results are checked for another 1,2 million primes.
The small numbers in the figure are rates of the nearby frequencies.
Shake 7b: There are no primes with Reduced Number 3, 6 or 9 besides 3 itself.
This speeds up prime sieving, but more important - ensures a subexponential
growth in prime numbers. This is a fact, not a fitting as the Riemann Hypothesis.
The music of the primes is a chord.
...but not of a circel groups most harmonic (with best packing ability) root-values - the triangulars 3, 6 and 9. That is why prime music sounds sour.
The music of the primes is a chord.
Primes are analysed in vertical batches of 125.000 as found here:
The result is given in the table below. Note that the significance of the result permeates all scales as the first million primes spans many (7) complete and fair rounds of First Digits (FD) 1-9. The result when restricting the amount of prime numbers to those with values <1.000.000 is shown in the small table.
Both the extraordinary 1,3,7,9 (primary respondents?) and the other digits show a Benfordian filling to the asymptotic uniform distribution like birthdays in Shake 1. If you have problems with seeing the Benfordian disrtribution, make frequency histograms for first, second digits etc. Several other examples on Benford`s Law (rules natural processes) with primary responses in mathematics and natural sciences are analysed in the book Karikaturer. One example on particle physics relates the Fine Structure Constant 1/137 (=0,0072992700...) with remarkable affinity to Arnold`s Triality Theorem, blow-ups in 27 Lines On a Cubic Surface, the mirror numbers 72 and 27 of root-values in Lie E-algebra and a proposed Circle-group based quantum mechanical model (briefly introduced in Shake 2) consistent with daily life large scale observations.
A numerological explanation of the Fine Structure Constant
would only be acceptable if it came from a more fundamental theory
that also provided a Platonic explanation of the value.
I. J. Good
The mystery about α is actually a double mystery.
The first mystery – the origin of its numerical value α ≈ 1/137 has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.
Malcolm H. Mac Gregor
The Benfordian distribution of the primary respondents 1, 3, 7 and 9 might be interpreted as quantized integrals of the prime function. To work this out analytically, my guess in Shake 7b below is to treat primes as a fluid dynamic througout the number system seen as discrete expanding pipelines.
For primes higher than the first million above, the First Digits stays much longer on 1 (Benfordian process). This gives rise to greater variations in the frequencies for 1, 3, 7 and 9, hence also in the dimensionless hypothesized fine structure constant - a fractal order in nature. Base 10 number system maps the order with zero skewness, giving rise to new types of "circadian rythms" beeing purely reflected in numbers (Shake 1). Hence digits corresponds to primary respondents and can possibly be intepreted as integrals of functional processes. And as you can see at the bottom on this page for the distribution of digits 0-9 within a prime - when primes gets higher, digit 9 increases in frequency. So the leading role iterates between 1s starting and 9s finishing/packing and shifts to 9 near(er) the asymptotic line - implications for the non-trivial zeros? Extreme or super hardpacking by 9 at asymptotic level will probably diminish the sub-exponentiality. This can be the reason why consecutive LD 1 are most common in the beginning (due to Benford`s Law) and consecutive LD 9 increases in the end (due to packing). Although Base 10 is the number system with zero skewness, I find it a bit troublesome to say that - what clockwise comes between 9 and 10 is nothing, but there is where 0 appears. It from bit..?
Shake 7a: Primes Last Digits 1, 3, 7 and 9 permeate throughout it`s whole complex;
being the most frequent digits, not uniformly rather Benfordian
- significant as the Fine Structure Constant.
....symbolizes the idea that every item of the physical world has at bottom — a very deep bottom, in most instances — an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-or-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is aparticipatory universe.
John A. Wheeler
Shake 7c: Frequency distribution for prime digits tends to be the opposite of Benfords Law (propulsion) the higher the prime is.
Note that all known Mersenne primes (besides the first) have Last Digits (propulsors) 1 and 7 only - indicating the rare occurance of these primes.
FD=First Digit, RN=Reduced Number, LD=Last Digit, SLD=Second Last Digit
The frequency table below shows for M-primes:
1) Alternating sequence in the distribution of
Second Last Digits with respect to Last Digits.
2) Reduced Number tends to be 1 and 4.
3) Reduced Number tends to be 4 when Last Digit is 1.
1) Purple colour indicate the propulsive effect of Last Digits as propulsors.
2) Primes Benfordian process is the propulsion.
3) There is much knowledge in Second Last Digits spike of 5. Generally (for high N); compare it to the similar spike of the midth of birthdays 1-31 in Go safer on the menu line. Specifically (allotted); for the birthdays of physics laureates in the book Karikaturer or in language. Comprehend..?
The table below shows a sum-up of the split-table above and bar diagrams of the two numerical prime principals in the case of M-primes.
...in the near future,
Shake 7d will give a single,
basic numerological and mathematcial explanation
for why primes Last Digits are 1, 3, 7 and 9.
It will be supported by new and extraordinary
numerical evidence, statistically significant.
What is your own opinion about why LD 1, 3, 7 and 9?
(Note that all strong pseudo-primes have RN 1 and factorises to x * y, both with RN 4 or 7.)
No sixes for propulsive work. Speculative explanation in Shake 7b, "Other observations and hypotheses".