What is the Visual Prime Pattern Based on Trig and Harmonics?

[Raphie]

Yes! When you add them together you get 7.47790847! Amazing!

Thank you for this extremely, uhh... "constructive" input Flexgunship. That 1 + 1 = 2 is also quite "amazing," but hardly (IMHO) worthy of an exclamation point, derisively intended or otherwise...

 

[JeremyEbert]

Raphie
Thanks again! I can see where my animation highlights the divisor function and its summation by counting the lattice points that intersect.

http://www.tubeglow.com/test/Fourier.swf

I'lll have the equation here shortly.

so if we go back to the y = sqrt((2xn)+n^2) parabolas. Another way to generate them would be:
d=(1,2,3,...n)
as y increases by sqrt(n)
x = (n-d^2)/(2d)
when x = 0, d = sqrt(n)

primes in mod(0.5) x = 0 when d = 1 or n

make sense?

 

[JeremyEbert]

a little more detail:
http://i98.photobucket.com/albums/l267/alienearcandy/prime-squarenewconic90-3d-2.png
could this be applied to spherical harmonics?

 

[JeremyEbert]

a little more detail:
http://i98.photobucket.com/albums/l267/alienearcandy/prime-squarenewconic90-3d-2.png
could this be applied to spherical harmonics?

Albrecht Durer's cone: http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2591&bodyId=3058

 

[JeremyEbert]

so if we go back to the y = sqrt((2xn)+n^2) parabolas.

the parabola sqrt((2xn)+n^2) is the same as sqrt(4 * (n/2) * (x + (n/2)) which I guess fits the more common definition of "y^2 = 4px".

Also Raphie, is this part clear?

d=(1,2,3,...n)
as y increases by sqrt(n)
x = (n-d^2)/(2d)
when x = 0, d = sqrt(n)

primes in mod(0.5) x = 0 when d = 1 or n

I've made reference to it before with little success.

 

[Raphie]

a little more detail:
http://i98.photobucket.com/albums/l267/alienearcandy/prime-squarenewconic90-3d-2.png
could this be applied to spherical harmonics?

Factorials, double factorials (product of odd numbers) and powers of 2 come into play in regards to Volumes of n-balls...

n-ball
http://en.wikipedia.org/wiki/N-sphere#n-ball

And so too Spherical Harmonics...

Hyperspherical volume element
http://en.wikipedia.org/wiki/N-sphere#Hyperspherical_volume_element

So... when summing volumes for a unit sphere, then e will naturally also be involved.

e.g.
pi^e/n! = SUM[V_2n]

Insofar as e relates to the prime number distribution [pi (x) ~ x/ln(x)] specifically, and compound growth generally, that there is some manner of relationship, discovered (and I am unaware of it) or undiscovered, seems evident. The precise nature of this relationship, however, is far less clear.

Keep in mind, however, that the number of conjugacy classes in the Symmetric Group S_n is a partition number:

Conjugacy class
http://en.wikipedia.org/wiki/Conjugacy_class

Since we now know, by the work of Ono et al, that partitions of prime numbers evidence fractal-like behavior, we can also logically surmise that the growth sequences of n-dimensional spaces of dimension p and/or p-1 (and/or p+1) will also be found to exhibit fractal-like behaviors. Think of it this way, and then the root system of a lattice such as E8 (241 is prime, and so too 239...) can, in some manner at least, be thought of as if it were a freeze-framed cross-section of a fractal iterating through multi-dimensional space.

And, insofar as all of this is the case, then Periodicity (e.g. The Crystal Restriction Theorem) and Quasi-periodicity (e.g. Penrose Tilings, related to the Golden Ratio) should also make an appearance is some form. (And so too, for that matter, the Shell theorem that you posted, which has everything to do with theoretical physics...)RF

As for this...

Also Raphie, is this part clear?.

I need to look more closely at what you've been doing before I can answer.

 

[Raphie]

Incidentally, Jeremy, if the train of thought I outlined above is on the right track, then relationships/mappings such as the following are most likely not coincidence...

For...

K --> Kissing Number (Lattice)
p --> Prime Number
B --> Bernoulli Number
5,7,11 --> Primes of the Ramanujan Partition Number Congruences (aka "0-D Ono Primes")

===================================

K_1 = 002 = 5^0 + 1 = 5^((05-5)/2) + 1 = 2^2 -2 = 2^(05-1)/2 - 2 = totient (p_2)
K_2 = 006 = 5^1 + 1 = 5^((07-5)/2) + 1 = 2^3 -2 = 2^(07-1)/2 - 2 = totient (p_4)
K_7 = 126 = 5^3 + 1 = 5^((11-5)/2) + 1 = 2^7 -2 = 2^(11-1)/2 - 2 = totient (p_31)

1 Division of 4-space --> 2 Regions
2 Divisions of 4-space --> 4 Regions
5 Divisions of 4-space --> 31 Regions
------------------------------------------
2*4*31 = 248 --> 240 + 8 = Dimensions of E_8

p_001 = p_(2^(05-1)/2 - 3) = 2
p_005 = p_(2^(07-1)/2 - 3) = 11
p_125 = p_(2^(11-1)/2 - 3) = 691
------------------------------------------
Partition Numbers: 1,1,2,3,5,7,11...

B_(05+1) = -1/30; 30 = 2*3*5
B_(07+1) = 1/42; 42 = 2*3*7
B_(11+1) = -691/2730 = 2*3*5*7*13
------------------------------------------
2,3,5 = ({5,7,11} - 1)/2 --> First 3 Sophie Germain Primes
pi (2,3,5,7,13) = 1,2,3,4,6 --> Allowable n-fold Rotational Symmetries under the Crystallographic Restriction Theorem

===================================

(p_(2^((5-1)/2) - 3) * p_(2^((7-1)/2) - 3) * p_(2^((11-1)/2) - 3) / ((5 * 7 * 11) * (B_(5+1) * (B_(7+1) * (B_(11+1))
(p_(5^((5-5)/2) - 0) * p_(5^((7-5)/2) - 0) * p_(5^((11-5)/2) - 0) / ((5 * 7 * 11) * (B_(5+1) * (B_(7+1) * (B_(11+1))
= (2*11*691)/(5*7*11*(-1/30 * 1/42 * - 691/2730))
= 196560

196560 --> Vertices of the Leech Lattice
= K_24

Bernoulli Numbers: Ramanujan's congruences
(Note: This set of Congruences is different from Ramanujan's Partition Number Congruences)
http://en.wikipedia.org/wiki/Bernoulli_number#Ramanujan.27s_congruences

Bernoulli Numbers: Restatement of the Riemann Hypothesis
http://en.wikipedia.org/wiki/Bernoulli_number#A_restatement_of_the_Riemann_hypothesis

196560/24 (Average Spheres/Dimension of the Leech Lattice), btw, = 8190, which is the 12th Ore's Harmonic Number and the totient of the 1028-th (=2^10 + 2^2) prime number. Also: sqrt ((196560*(-1/30 * 1/42 * - 691/2730))) gives a not very accurate (pi is 99.9907% of this...), but interesting in a contextual sense, approximation of 2*pi.

And sqrt ((196560*(-1/30 * 1/42 * - 691/2730))), by extension, gives a not very accurate approximation of 4*pi^2, which, of course, prominently figures into Kepler's Third Law, perhaps of interest to you in relation to the Shell Theorem.RF

 

[Raphie]

Just a thought for you Jeremy, but if you're looking to relate the primes to gravity in the exploratory vein you seem to be doing, then try substituting (Zeta_2)^-1 = 6/pi^2 into the formula for a pendulum (replacing "L/g"); Zeta_2, of course, being the probability that two randomly selected numbers will be relatively prime. When you square the period, you get the result: 24 seconds^2.

All primes > 3 when squared, as you may already know, are == 1 mod (24), while ALL odd primes are congruent to either 1 mod (24), or 1 mod (totient 24) = 1 mod (8) .

Also, (p^2 - 1)/24 is pentagonal for n > 3. Thus, lim n --> infinity p^2/T^2 is a Pentagonal "Number," as well as 1/3 a Triangular "Number," at least conceptually speaking (since infinity is not a "number"). Add in a distance parameter to the numerator (p^2 meters^2 - 1^2 meter^2)/T^2 and you end up with velocity^2 unit-wise.

If you plug in Zeta_4, you also get some interesting results related to the steradian, 1/90-th of a steradian, to be specific (totient 90 = 24); the steradian also being known as a "solid angle." 4*pi Steradian (= 4*pi*radians^2) traces the surface area of a sphere.RF

RELATED LINK
John Baez
The Rankin Lectures, University of Glasgow
September 15-19, 2008
My Favorite Numbers (5, 8 & 24)
http://math.ucr.edu/home/baez/numbers/

Also see the discussion on the below referenced thread. There's a great link there, posted by (former poster) Goongyae, to a Euler paper that relates generalized pentagonal numbers to the primes...
Ken Ono and Factoring?
https://www.physicsforums.com/showthread.php?t=472486

 

[JeremyEbert]

Raphie,
Thanks for the wealth of information. Your posts really help me to see things I've never thought about before. Sorry for the delay but I've been trying to formalize my ideas before I get to lost exploring other areas. I have attached a rough draft of what I've been working on. I'd love any feedback you can offer.
Jeremy

 

[JeremyEbert]

I fixed a few mistakes.
http://www.tubeglow.com/test/Pythagorean lattice.pdf

 

[Anymodal]

i wish i was smart

amazing article and amazing animtion
I find so infinetly interesting... can you explain me shortly the animation? i don't really understand what it means. I am doing 1st year eng. so i don't know that much math

 

[Raphie]

I fixed a few mistakes.
http://www.tubeglow.com/test/Pythagorean lattice.pdf

I haven't had the time to properly respond Jeremy, but I just want you to know I believe, in general, you are on a promising track with your explorations. Check out the Statistics version of the Pythagorean Theorem... Variance. A + B = C.

VAR(X) = E[X]^2 - E[X^2]

Also note the following symmetrical equation form: x^2 + 2xy + y^2 = z.

RELATED LINKS:

Variance
http://en.wikipedia.org/wiki/Variance

The Expectation Operator
http://arnoldkling.com/apstats/expect.html
E(X+Y)^2 = E(X^2 + Y^2 + 2XY)

Expected value
http://en.wikipedia.org/wiki/Expected_valueRF

 

[JeremyEbert]

very interesting patterns if I use a little recursion. Its in 3D best viewed in 1900*1200 resolution. Keep the mouse off of the page while it loads and it will plot in 2d first. source code is available. http://www.tubeglow.com/test/PL3D/P_Lattice_3D.html

 

[Raphie]

very interesting patterns if I use a little recursion. Its in 3D best viewed in 1900*1200 resolution. Keep the mouse off of the page while it loads and it will plot in 2d first. source code is available. http://www.tubeglow.com/test/PL3D/P_Lattice_3D.html

I have to say, Jeremy: Super cool. And, I might add, your 3-D graphic does little to dissuade me from suspecting that a fractal geometry comes into play not just in relation to primes and partition numbers, but also in relation to the positioning of the primes.

Keep in mind, ala Marcus du Sautoy, that every time you add in a new prime you are, in effect, adding in another variable that will create "waves" and "ripples" interacting with all the other waves and ripples created by the other primes, meaning that the shapes you model will morph endlessly as you go further and further down the number line, or, rather, further out from the origin along the surface of your Minkowski-esque "prime lattice light cone."

But, just because you know that it will "morph," this in no way excludes the possibility, even perhaps likelihood, of regularities in relation to the manner of "timing" by which new variables are introduced, because each and every new variable is recursively made possible only by the "multiplicative failure" of the primes that preceded it to fully "cover" "number space".

Also keep in mind that variables equate with dimensions:

Variable == Parameter == DimensionRF

 

[JeremyEbert]

But, just because you know that it will "morph," this in no way excludes the possibility, even perhaps likelihood, of regularities in relation to the manner of "timing" by which new variables are introduced, because each and every new variable is recursively made possible only by the "multiplicative failure" of the primes that preceded it to fully "cover" "number space".

I think the "timing" is based on natural squares. Also, incase you didn't notice the conic section from the fractal. http://i98.photobucket.com/albums/l267/alienearcandy/Conic.png

edit:
I attached a better image and its projection. this is a kind of root system right?
http://en.wikipedia.org/wiki/Root_system
http://upload.wikimedia.org/wikiped...vg/1000px-Integrality_of_root_systems.svg.png

 

[JeremyEbert]

More "timing" links. Natural squares and partition numbers maybe? this is based on 12:
http://1.bp.blogspot.com/_u6-6d4_gsSY/TESnJ1Q3w8I/AAAAAAAAACk/vwiVbGAzz1Y/s1600/roots.PNG

 

[Raphie]

More "timing" links. Natural squares and partition numbers maybe? this is based on 12:
http://1.bp.blogspot.com/_u6-6d4_gsSY/TESnJ1Q3w8I/AAAAAAAAACk/vwiVbGAzz1Y/s1600/roots.PNG

Offhand, seems to me you should also think about considering natural cubes. In other words, not just squares and not just cubes, but both. Meaning, you might want to familiarize yourself with the Eisenstein integers.

And in relation to the number 12...

n(n+ceiling(2^n/12))
http://oeis.org/A029929

The first 8, but only 8 numbers in this series are proven (lattice) kissing numbers.

- RF

RELATED LINK
Leech lattice
http://en.wikipedia.org/wiki/Leech_lattice
The Leech lattice is also a 12-dimensional lattice over the Eisenstein integers. This is known as the complex Leech lattice, and is isomorphic to the 24-dimensional real Leech lattice...

 

[Raphie]

Offhand, seems to me you should also think about considering natural cubes. In other words, not just squares and not just cubes, but both. Meaning, you might want to familiarize yourself with the Eisenstein integers.

Subtract any two + and - Pentagonal Pyramid numbers of equal index and you get a square. Add them together and you get a cube.

e.g.
40 - 24 = 4^2
40 + 24 = 4^3

One can use this mathematical fact to easily obtain integer solutions to the following:

Period^2 = 4*pi^2/GM * Distance^3 (Kepler's 3rd Law)

e.g.
(40 + 24)^2 = (40 - 24)^3 = 4^6 = 4096
(6 + 2)^2 = (6 - 2)^3 = 2^6 = 64 (= sqrt 4096)

The Pentagonal Pyramid numbers, of course, are the summation of the Pentagonal numbers, which are already well-known to be related to the "timing" and/or "tuning" of the primes.

p^2 - 1 == 1 mod (24) for all p > 3

(p^2 - 1)/24 is Pentagonal for all p > 3.

And, also, as I mentioned previously, 24 s^2 is the Period^2 one obtains if one replaces L/g in the formula for a pendulum with zeta(2)^-2 = (pi^2/6)^-2, where (the reciprocal of) zeta(2) gives the probability of two randomly selected integers being relatively prime.

- RF

Note: Pentagonal Pyramid Numbers have a very easy to remember formula n*T_n = (+) Pentagonal Pyramid # and n*T_-n = (-) Pentagonal Pyramid #, for T_n a Triangular Number.

 

[JeremyEbert]

Raphie,
Thanks! I'm working on some projections.I definately see an Eisenstein integer connection.

edit: I'm working on the pendulum

 

[Raphie]

n(n+ceiling(2^n/12))
http://oeis.org/A029929

Jeremy, in regards to the formula up there above, I just want to give you one tiny little example of the kind of relationships I am finding that have me going "hmmmm..." in relation to the sequencing of the primes...

A060967
Number of prime squares <= 2^n.
http://oeis.org/A060967
0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 24...

RULE: SUBTRACT 1
-1, -1, 0, 0, 1, 2, 3, 4, 5, 9, 10, 13, 17, 23...

RULE: ITERATE INTO THE "QRIME" NUMBER SEQUENCE {0, 1 U Primes} indexed from -1; p'_(n-1)
0, 0, 1, 1, 2, 3, 5, 7, 11, 17, 29, 41, 59, 83...

RULE: ADD 1
1, 1, 2, 2, 3, 04, 06, 08, 12, 018, 030, 042, 060, 084...

RULE: |MULTIPLY BY (n-2)|
2, 1, 0, 2, 6, 12, 24, 40, 72, 126, 240, 378, 600, 924...

In formula form...
K_(n-2) = (n-2) * (1 + p'_(-1 + COUNT[Number of prime squares <= 2^n])) for n = 2 --> 10

The 3rd through 11th values are the (proven lattice) Kissing Numbers up to Dimension 8, the very same ones you get by inserting n into the formula: n(n+ceiling(2^n/12)). And the 2nd and 12th values? T_1^3 = 1 and T_3^3 = 378 (& 600 = 2*T_24, while 924 is a Central Binomial Coefficient, the sum of proper divisors of which = 1764 == 42^2)

378 - totient (107) = 272 = K_9; 107 = p'_28 = p'_(T_7) = p'_(T_(Lucas_4)
001 - totient (002) = 000 = K_0; 002 = p'_01 = p'_(T_1) = p'_(T_(Lucas_1)

2 is the 1st Mersenne Prime Exponent, and 107 the 11th (1 = Lucas_1, 11 = Lucas_5, and 1 and 28 are both k-Perfect Numbers). These two numbers also have the property, that, when triangulated, you get a Lucas Number. There are only 3 Triangular Lucas Numbers: 1, 3 and 5778. (See: Lucas Number http://mathworld.wolfram.com/LucasNumber.html )

0001 = T_001 = Lucas_01 = Lucas_(Lucas_1) = Lucas_(Lucas_(T_1)) = Lucas_(1*T_1)
5778 = T_107 = Lucas_18 = Lucas_(Lucas_6) = Lucas_(Lucas_(T_3)) = Lucas_(2*T_2)

(1 and 6 are k-Perfect Numbers, 1,6 & 18 are the first 3 Pentagonal Pyramid Numbers)

Also, see...
k*Lucas_n + 1 is a prime of Lucas Number Index
https://www.physicsforums.com/showthread.php?t=497766

- RF

 

[Raphie]

Jeremy, in regards to the formula up there above, I just want to give you one tiny little example of the kind of relationships I am finding that have me going "hmmmm..." in relation to the sequencing of the primes...

A060967
Number of prime squares <= 2^n.
http://oeis.org/A060967
0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 24...

RULE: SUBTRACT 1
-1, -1, 0, 0, 1, 2, 3, 4, 5, 9, 10, 13, 17, 23...

RULE: ITERATE INTO THE "QRIME" NUMBER SEQUENCE {0, 1 U Primes} indexed from -1; p'_(n-1)
0, 0, 1, 1, 2, 3, 5, 7, 11, 17, 29, 41, 59, 83...

RULE: ADD 1
1, 1, 2, 2, 3, 04, 06, 08, 12, 018, 030, 042, 060, 084...

RULE: |MULTIPLY BY (n-2)|
2, 1, 0, 2, 6, 12, 24, 40, 72, 126, 240, 378, 600, 924...

In formula form...
K_(n-2) = (n-2) * (1 + p'_(-1 + COUNT[Number of prime squares <= 2^n])) for n = 2 --> 10

The 3rd through 11th values are the (proven lattice) Kissing Numbers up to Dimension 8, the very same ones you get by inserting n into the formula: n(n+ceiling(2^n/12)).

A POSSIBLY RELATED SEQUENCE
Suppose the sum of the digits of prime(n) and prime(n+1) divides prime(n) + prime(n+1). Sequence gives prime(n).
http://oeis.org/A127272
2, 3, 5, 7, 11, 17, 29, 41, 43, 71, 79, 97, 101, 107...

e.g.
(2 + 3)/(2+3) = 1
(3+5)/(3+5) = 1
(5+7)/(5+7) = 1
(7+11)/(7+(1+1)) = 2
(11+13/((1+1)+(1+3)) = 4
(17+19/((1+7)+(1+9)) = 2
(29+31/((2+9)+(3+1)) = 4
(41+43/((4+1) + (4+3)) = 7
(43+47/((4+3)+(4+7)) = 5
(71+73)/((7+1)+(7+3)) = 8
(79+83)/((7+9)+(9+7)) = 5
(97+101)/((9+7)+(1+0+1)) = 11
(101+103)/((1+0+1) + (1+0+3) = 34
(107+109)/((1+0+7)+(1+0+9) = 12

ALSO...
Numbers n such that 1 plus the sum of the first n primes is divisible by n+1.
http://oeis.org/A158682
2, 6, 224, 486, 734, 50046, 142834, 170208, 249654, 316585342, 374788042, 2460457826, 2803329304, 6860334656, 65397031524, 78658228038

002 - 002 = 000 = K_00
012 - 006 = 006 = K_02 (Max)
600 - 224 = 336 = K_10 (Lattice Max known)
924 - 486 = 438 = K_11 (Lattice Max known)

6/(5+1) = 1
42/(6+1) = 6
143100/(224+1) = 636
775304/(486+1) = 1592

Like I said, especially given that these two progressions are ones I came across in the process of writing that last post to you, "hmmmm..."

RELATED PROGRESSIONS
Integer averages of first n noncomposites for some n.
http://oeis.org/A179860
1, 2, 6, 636, 1592, 2574, 292656, 917042, 1108972, 1678508, 3334890730, 3981285760, 28567166356, 32739591796, 83332116034

a(n) is the sum of the first A179859(n) noncomposites.
http://oeis.org/A179861
1, 6, 42, 143100, 775304, 1891890, 14646554832, 130985694070, 188757015148, 419047914740, 1055777525624570390, 1492138298614167680, 70288308055831268412, 91779857115464381780, 571686203669195590338

Numbers n that divide the sum of the first n noncomposites.
http://oeis.org/A179859
1, 3, 7, 225, 487, 735, 50047, 142835, 170209, 249655, 316585343, 374788043, 2460457827, 2803329305, 6860334657

This number, in particular, I find interesting...
142835 = 5*7^2*11*53 = (142857 - par_8) = (142857 - 22)
vs. 1/7 = .142857 (repeating)
Indexing from 0, 142857 is the 24th Kaprekar Number

1, 3, 7 and 225, the 1st 4 terms in that last sequence above == (2^1 - 1)^1, (2^2 - 1)^1, (2^3 - 1)^1, (2^4 - 1)^2.

- RF

 

[Raphie]

Jeremy, as an FYI, and by way of giving another example, if one desires to mathematically derive, say, the Dimension 10 Lattice Kissing Number from a convolution of primes and partition numbers, a far simpler way to do it is as follows:

p'_((par_n - 1) * p'_(par_(n-1) - 1)

p'_(1-1) -1 = 0
p'_(1-1) -1 = 0
p'_(2-1) -1 = 1
p'_(3-1) -1 = 2
p'_(5-1) -1 = 6
p'_(7-1) -1 = 12
p'_(11-1) -1 = 28

0*0 = 0 = K_0
0*1 = 0 = K_0
1*2 = 2 = K_1
2*6 = 12 = K_3
6*12 = 72 = K_6
12*28 = 336 = K_10

In order, that formula returns maximal (proven except for Dimension 10) lattice sphere packings for Dimensions equal to 6 consecutive Triangular Numbers:

T_-1, T_0, T_1, T_2, T_3, T_4

On the other hand, if you simply add 1 to the first 7 partition numbers, and multiply by n...

(1+1)*0 = 0 = K_0
(1+1)*1 = 2 = K_1
(2+1)*2 = 6 = K_2
(3+1)*3 = 12 = K_3
(5+1)*4 = 24 = K_4
(7+1)*5 = 40 = K_5
(11+1)*6 = 72 = K_6

... then you get Maximal (proven) Lattice Sphere packings to dimension 6.RF

 

[JeremyEbert]

I'm going through you posts now. I reworked the visual a little. Click any where on the page after it loads the black back ground then press:

1 = normal growth of the equation. after it builds for a while you can notice the pattern and the timing. Seems to be timed like a pendulum.

or

2 = normal "inverse growth".

or

3 = fractal pattern generationup/down arrrow = zoom in out

left/right arrow = fractal limit increase/decrease.

d = 3d on/off

http://www.tubeglow.com/test/PL3D2/P_Lattice_3D_2.html

 

[Raphie]

Jeremy, firstly, the page you linked to doesn't seem to work with my system.

Secondly, I wouldn't read too much into any single example I might give. It's all of the examples, taken together, and the picture they are seeming to paint (or the tune they are seeming to play) that I find most interesting.

Thirdly, a critic would reasonably note that the indices of the prime numbers I am giving are all quite small. And that's a fair point. But then one has to explain away as "coincidence" relationships such as the following:

For 1, 2, 3, 4 and 6 the solutions to the Crystallographic Restriction Theorem, then consider lattices in the following Dimensions:

(1 - 1)^2 + 1 - totient (1) = 0
(2 - 1)^2 + 1 - totient (2) = 1
(3 - 1)^2 + 1 - totient (3) = 3
(4 - 1)^2 + 1 - totient (4) = 8
(6 - 1)^2 + 1 - totient (6) = 24
Dimensions {0 & 24} Union {1, 3, 8}, the dimensions associated with the Standard Model of Physics = SU(3)×SU(2)×U(1)

Then, for F_n a Fibonacci Number and T_n a Triangular Number...

And for...

2, 4, 6, 10, 22 == totient (1st 5 safe "qrimes") == 2 * (1, 2, 3, 5, 11) where...

01 = p'_(1 - 1) = par_1
02 = p'_(2 - 1) = par_2
03 = p'_(3 - 1) = par_3
05 = p'_(4 - 1) = par_4
11 = p'_(6 - 1) = par_6

Then...
p_00001 - p_01 = p_F_02 - p_((F_0)*(T_(pi(pi(01) + 1))) + 1) = 000002 - 002 = 000002 = K_0
p_00003 - p_02 = p_F_04 - p_((F_1)*(T_(pi(pi(02) + 1))) + 1) = 000005 - 003 = 000002 = K_1
p_00008 - p_04 = p_F_06 - p_((F_2)*(T_(pi(pi(03) + 1))) + 1) = 000019 - 007 = 000012 = K_3
p_00055 - p_07 = p_F_10 - p_((F_3)*(T_(pi(pi(05) + 1))) + 1) = 000257 - 017 = 000240 = K_8
p_17711 - p_31 = p_F_22 - p_((F_4)*(T_(pi(pi(13) + 1))) + 1) = 196687 - 127 = 196560 = K_24

Note: 1, 2, 3, 5 & 13 are the Prime Numbers | (2^n - 1) is Twice Triangular (aka "The Ramanujan-Nagell Pronic Numbers"). And 2, 3, 5, 17 and 257 are all Fermat Primes, while 2, 3, 7, 17 and 127 (and also 19) are all Mersenne Prime Exponents, the 1st, 2nd, 4th, 6th and 12th (19 is the 7th).

p'_1 - 1 = 02 - 1 = 1
p'_2 - 1 = 03 - 1 = 2
p'_3 - 1 = 05 - 1 = 4
p'_4 - 1 = 07 - 1 = 6
p'_6 - 1 = 13 - 1 = 12

The condensed way to state the above is as follows:
--------------------------------------------------------------------------------
for...
K_n = n-th Kissing Number
p'_(n-1) = n-th n in N | -1 < d(n) < 3 --> {0,1,2,3,5,7,11,13...}
c_(n-1) = n-th n in N | -1 < totient(n) < 3 --> {0,1,2,3,4,6}
E_n = n-th Mersenne Prime Exponent
F_n = n-th Fibonacci Number
then for range n = 0 --> 4...

FORMULA
K_((c - 1)^2 + 1 - totient (c))
=
(p'_(F_(2(p'_(c - 1))))) - (E_(p'_c - 1))
--------------------------------------------------------------------------------

2, 3, 5, 7, 13 [= {p_c} == {n in N | d(p_c - 1) = c}], as well as being the first 5 Mersenne Prime Exponents, are also the unique prime divisors of the Leech Lattice: K_24 = 196560

And, as I believe you may already know, this particular set of primes has been associated with anomaly cancellations in 26 Dimensional Bosonic String Theory by Frampton and Kephart:

Mersenne Primes, Polygonal Anomalies and String Theory Classification
http://arxiv.org/abs/hep-th/9904212RF

============================================
Also...

00 = p'_-1 = (1 - 1)
------------------------
01 = p'_(p'_-1) = (2 - 1)
02 = p'_(p'_(p'_-1)) = (3 - 1)
03 = p'_(p'_(p'_(p'_-1))) = (4 - 1)
05 = p'_(p'_(p'_(p'_(p'_-1)))) = (6 - 1)
11 = p'_(p'_(p'_(p'_(p'_(p'_-1))))) = (12 - 1)
for 1, 2, 3, 4, 6, 12 --> the divisors of 12

And also...

(01 * 0) + 3 - d(01) = 02 = p'_01 --> 01st Mersenne Prime Exponent
(02 * 2) + 3 - d(02) = 05 = p'_03 --> 03rd Mersenne Prime Exponent
(03 * 4) + 3 - d(03) = 13 = p'_06 --> 05th Mersenne Prime Exponent
(05 * 6) + 3 - d(05) = 31 = p'_11 --> 08th Mersenne Prime Exponent
(11 * 8) + 3 - d(11) = 89 = p'_24 --> 10th Mersenne Prime Exponent
for 1, 3, 5, 8, 10 --> Sum of Divisors (SUM d(n)) for n = 1 through 5

Here are the 1st 14 Mersenne Prime Exponents (inclusive of 1)...
1, 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521 (Range = Lucas_1 --> Lucas_13)

As you may or may not have noticed, in the last few posts I've referenced every one of these excepting 61 and 521 (= Lucas_13 = Lucas_(p'_(sigma_5)), indexing from 0, the 13th Mersenne Prime Exponent). ((61-1)*11) = totient (p'_11^2) = 660 = (T_36 - sqrt (36)), by the way, is a simple group that, musically speaking, is one-perfect 5th above A-440 and its my prediction for the maximal Kissing Number in 11 dimensions (+ or - 12). "Coincidentally," 660 - 12 = 648, is the maximal known lattice sphere packing in 12 dimensions (= 2*18^2 = Lucas_0*Lucas_(sigma_(5))^2 = p'_(Lucas_5)^2 - p'_(sigma_5)), while 36 (=2*18) is the totient of the 12th prime number, 37.

Finally, in the interests of clarity, note that the below are all just absurdly long-winded, even if contextually relevant, ways of stating: 0, 1, 2, 2, 4:

(pi(pi(01) + 1)) == d(pi (01)); 01 --> (1-1)th Mersenne Prime Exp == pi (pi (02)); 02 --> 1st Mersenne Prime Exp = 0
(pi(pi(02) + 1)) == d(pi (02)); 02 --> (2-1)th Mersenne Prime Exp == pi (pi (03)); 03 --> 2nd Mersenne Prime Exp = 1
(pi(pi(03) + 1)) == d(pi (03)); 03 --> (3-1)th Mersenne Prime Exp == pi (pi (05)); 05 --> 3rd Mersenne Prime Exp = 2
(pi(pi(05) + 1)) == d(pi (05)); 05 --> (4-1)th Mersenne Prime Exp == pi (pi (07)); 07 --> 4th Mersenne Prime Exp = 2
(pi(pi(13) + 1)) == d(pi (13)); 13 --> (6-1)th Mersenne Prime Exp == pi (pi (17)); 17 --> 6th Mersenne Prime Exp = 4

And finally, finally... a statement such as p_((F_4)*(T_(pi(pi(13) + 1))) + 1) is an even more long-winded way of stating: p_31, which is the 12th Mersenne Prime Exponent and/or the iterated 8th Mersenne Prime Exponent. As such, I went back in the post and included the condensed formula...

 

[Raphie]

The condensed way to state the above is as follows:
--------------------------------------------------------------------------------
for...
K_n = n-th Kissing Number
p'_(n-1) = n-th n in N | -1 < d(n) < 3 --> {0,1,2,3,5,7,11,13...}
c_(n-1) = n-th n in N | -1 < totient(n) < 3 --> {0,1,2,3,4,6}
E_n = n-th Mersenne Prime Exponent
F_n = n-th Fibonacci Number
then for range n = 0 --> 4...

FORMULA
K_((c - 1)^2 + 1 - totient (c))
=
(p'_(F_(2(p'_(c - 1))))) - (E_(p'_c - 1))
--------------------------------------------------------------------------------

All that said, Jeremy, again, recognize that one can make essentially the same statement far more simply...

for...
c_(n-1) --> {0,1,2,3,4,6} == pi {1, 2, 3, 5, 7, 13}
--> Divisors of Prime Divisors of Leech Lattice
--> Integers with Totient < 3
--> n in N | d(p_c - 1) = c
--> Solutions to 2*cos (2*pi/(n + 1 - sgn(n)) is in N
E_(n-1) --> {0,1,2,3,5,7,13,17,19...}
--> 0, 1 Union Mersenne Prime Exponents

then...
K_((c_n - 1)^2 + 1 - totient (c_n))
=
(2^E_(c_n - 1)) - 2)*((E_(c_n - 1) - 1)*totient (c_(n-1)))

Expansion:
K_00 = (2^01 - 2) * ((01 - 1)*totient (0)) = 0000 * 00 = 0
K_01 = (2^02 - 2) * ((02 - 1)*totient (1)) = 0002 * 01 = 2
K_03 = (2^03 - 2) * ((03 - 1)*totient (2)) = 0006 * 02 = 12
K_08 = (2^05 - 2) * ((05 - 1)*totient (3)) = 0030 * 08 = 240
K_24 = (2^13 - 2) * ((13 - 1)*totient (4)) = 8190 * 24 = 196560

---------------------------------------------------------------------------------
POSSIBLY RELATED PROGRESSION
y such that y^2=C(x,0)+C(x,1)+C(x,2)+C(x,3) is soluble
0, 1, 2, 8, 24, 260, 8672
R. K. Guy, Unsolved Problems in Number Theory, Section D3.
http://oeis.org/A047695
---------------------------------------------------------------------------------

c_n & E_(c_n - 1) - 1) can be linked in the following manner:

Denote i-phi(x)_n as the n-th integers with a totient of x (The "Inverted Totient Function")
Denote s(x) as the number of Solutions to i-phi(x)
Denote J_n as the y solutions to 2^y - 1 is Triangular (Ramanujan-Nagell Triangular Numbers)

Then...

i-phi(J) -->
------------------
i-phi(00) --> 00; ------- Solutions = 1 (Mathematica Definition)
i-phi(01) --> 01 02; ------- Solutions = 2
i-phi(02) --> 03 04 06; ------- Solutions = 3
i-phi(04) --> 05 08 10 12; ------- Solutions = 4
i-phi(12) --> 13 21 26 28 36 42; ------- Solutions = 6

00 = 2T_pi(01) = 2T_d(01-1) = 2T_0
02 = 2T_pi(02) = 2T_d(02-1) = 2T_1
06 = 2T_pi(03) = 2T_d(03-1) = 2T_2
12 = 2T_pi(05) = 2T_d(05-1) = 2T_3
42 = 2T_pi(13) = 2T_d(13-1) = 2T_6

And right there you've got yourself, potentially, a nice clean bridge between (just for starters...), Ramanujan-Nagell, the Solutions to the Crystallographic Restriction Theorem and the Divisors of the Leech Lattice/Frampton-Kephart Primes.

s(J) = c
i-phi(J)_1 = (J-1) + totient (c)
i-phi(J)_c = 2T_(d(J))
i-phi(J)_c = 2T_(pi(J+1))
and...
Delta (i-phi(J)_1, i-phi(J)_c) = p'_(J - 2) = 0, 1, 3, 7, 29

Thus, for instance...

K_(totient(s(J))J) = (2^J + 1)*(totient(s(J))J)

K_00 = (2^(00+1) - 2) * (00*1) = 0
K_01 = (2^(01+1) - 2) * (01*1) = 2
K_04 = (2^(02+1) - 2) * (02*2) = 24
K_08 = (2^(04+1) - 2) * (04*2) = 240
K_24 = (2^(12+1) - 2) * (12*2) = 196560

Hard to get more simple than that. With the nice little bonus that...

pi (2^01) = 0001 = 2^00 + 0 = 2^(T_-1) + 0
pi (2^02) = 0002 = 2^00 + 1 = 2^(T_0) + 1
pi (2^03) = 0004 = 2^01 + 2 = 2^(T_1) + 2
pi (2^05) = 0011 = 2^03 + 3 = 2^(T_2) + 3
pi (2^13) = 1028 = 2^10 + 4 = 2^(T_4) + 4

Thus...

pi (2^J+1) = 2^(T_(s(J) - 2)) + n

If that and the other relationships presented in this post don't make you and/or anyone else who comes across this go "hmmmm," then I really don't know what will.

Go back to the beginning of this post and you'll see you can make both formulas relating to the previous post far, far simpler by substitution. But it still doesn't change the nature of the relationships. Rather, all that changes is the apparent simplicity of the relationships.

e.g.

K_(n(mod 5)+0) = (s(J)_(n-1) + 0*6)(n+0)^0 = 00, 02, 06, 012, 024
K_(n(mod 5)+4) = (s(J)_(n-1) + 1*6)(n+4)^1 = 24, 40, 72, 126, 240RF

 

[Raphie]

In relation to the Richard Guy sequence posted above, ...

0^2, 1^2, 2^2, 8^2, 24^2, 260^2, 8672^2
= 0, 1, 4, 64, 576, 67600, 75203584

These squares correspond with Cake Numbers of index -1, 0, 2, 7, 15, 74 & 767

Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1.
http://oeis.org/A000125
[0], 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235, 15226...

I know this set of numbers well...

e.g.
Product [|Cake_n - 0|!/|Cake_n - 1|!] = 1, 1, 2, 8, 64, 960, 24960, 1048320, 67092480...

00000000!/|00000000 - 1|! = |0000^2 - d(0000) + 1|; 0000 = M_00
00000001!/|00000001 - 1|! = |0001^2 - d(0001) + 1|; 0001 = M_01
00000008!/|00000008 - 1|! = |0003^2 - d(0003) + 1|; 0003 = M_02
00000960!/|00000960 - 1|! = |0031^2 - d(0031) + 1|; 0031 = M_05
67092480!/|67092480 - 1|! = |8191^2 - d(8191) + 1|; 8191 = M_13

- RF

 

[Raphie]

Jeremy, I just mentioned the number 67092480. I mention it because you are looking at pendulums in relation to the primes and you're interested in gravity. 67092480 = (90^2 + 90)^2 + 2*(90^2 + 90) [follows the form of a parabola] is one of the key terms in the below formula which involves just one sign change to go from A to B. If only to protect myself from charges of "numerology," think of it as "sudoku," not physics, because the numbers are all based on lattices (and primes) and on the very "what if" proposition of "what if fractals were to guide the evolution of all dynamical systems, organic and inorganic alike, across all levels of organization from the very, very small to the very, very large (i.e. what if it were possible to develop an atomic model of the solar system as opposed to a planetary model of the atom?...)

For...
s(1 + 2 + 3 + 5 + 13) = s(24) = 10
Solutions i-phi(24) = 35, 39, 45, 52, 56, 70, 72, 78, 84, 90
Range: 55 = T_10 = F_10 = ceiling [e^((10-2)/2)]

A) The Gravitational Constant
((((4*pi^2)^(1-(-1)^0)*pi)/0.007297352570631)((2*10^34/10^(12 + (9*0)))*(35 + sqrt (35)/10^4))/(((67092480 - 1)^(2+0)/sqrt (67092480 + 1)^(2+0))*(299800649 - sqrt (67092480 + 1))))^(1/-1)
= 6.67428281 * 10^-11

B) The ~ Planetary Positioning Ratio @ n = 10
((((4*pi^2)^(1-(-1)^1)*pi)/0.007297352570631)((2*10^34/10^(12 + (9*1)))*(35 + sqrt (35)/10^4))/(((67092480 - 1)^(2+1)/sqrt (67092480 + 1)^(2+1))*(299800649 - sqrt (67092480 + 1))))^(1/2)
= 1.68845301

EMPIRICAL PLANETARY POSITIONING RATIO @ n = 10 Source for Values: Wikipedia Planet Pages
(with Asteroid Belt set at 2.816 AU, which yields minimal possible value)
((394.8165/301.0366) + (301.0366/192.2941) + (192.2941/ 95.8202) + (95.8202/52.0427) + (52.0427/28.16) + (28.16/15.2368) + (15.2368/10) + (10/7.2333) + (7.2333/3.8710))/ 9
~ 1.68845075

39.48165 + 30.10366 + 19.22941 + 9.58202 + 5.20427 + 2.816 + 1.52368 + 10 + .72333 + .38710 = 110.05112
floor[110.05112] = 2*T_10 = 2*55

The numbers in that equation are not at all "random." For instance...

299800649
= (G_(24-0) * L_(24+0))/10^2 ~ (G_(24-6) * L_(24+6))/10^2
= (289154*103682)/100 ~ (16114 * 1860498)/10^2
for G the Golden Scale (sum of 5 consecutive Fibonacci Numbers) and L the Lucas Series

Delta ((G_(n-0) * L_(n+0)), (G_(n-6) * L_(n+6))) = 256 == totient(p_55) == totient(p_(T_10))
6 == K_2 == L for pendulum set to Zeta(2)^-1 = L/g
24 ==K_4 == T^2 for pendulum set to Zeta(2)^-1 = L/g

(67092480 - 1)/sqrt (67092480 + 1)
= 8190.99976...
(10^34*(9.10938215*10^-31))/(10^34*(6.626067758602965*10^-34/(2*pi)))^2
= 8190.99976...

floor [299800649 - 8190.99976] = 299792458RF

Note:
72-45 = i-phi(24)_7 - i-phi(24)_3 = 27. 27 and 45 are the roots of E_6
56+70 = i-phi(24)_5 + i-phi(24)_6 = 126 = K_7. These are the roots of E_7...
A-D-E Classification
http://en.wikipedia.org/wiki/ADE_classification

RELATED LINK
A Cute Formula for Pi
https://www.physicsforums.com/showthread.php?t=475539

 

[JeremyEbert]

Raphie,
Your “what if” fractal scenario is exactly what I’ve been thinking for a while. To me, your posts have shown some amazing connections, I only wish I had the knowledge and insight to give you meaningful feedback other than, “ Holy S@#! Yea, I see the connection now!” I feel like I’m in a crash course in number theory and I love it, I only wish I could contribute more. I’m still reviewing the depth of your latest posts.
I think the link I posted was down for a while. It seems to be working on several different systems now. I would really like you to take a look at it if it’s working for you. As regards the “pendulums in relation to the primes” and my interest in gravity, I see some interesting results I think. After the page loads the black background, press “1” and let that run for 10 sec or so, a pendulum type motion starts to become apparent. Now press “2”. It “flips” the equation so it’s contracting instead of expanding. It resembles, to me, a “ball in a cone” . Other functions again:
1 = Expansion
2 = Contraction
3 = Recursion (Reload page first)
UP/DOWN = ZOOM IN/OUT
LEFT/RIGHT = RECURSION LIMIT. Default limit of 2 is loaded. Increasing the limit yields some familiar patterns.
d = 3D ON/OFF

http://www.tubeglow.com/test/PL3D2/P_Lattice_3D_2.html

 

[Raphie]

I think the link I posted was down for a while. It seems to be working on several different systems now.
http://www.tubeglow.com/test/PL3D2/P_Lattice_3D_2.html

Sorry, Jeremy, but it's still not working for me. I use a Mac that's pretty buggy. For instance, the Physics Forums latex code generator doesn't work for me either.

As for crash courses in "number theory." I wouldn't really call it that. If anything, you're getting a crash course in applied social theory.

The little ditty that guides many of my mathematical explorations?

Where e and F ride side by side, the orbits may not collide.
Where they diverge clear order may not emerge.

But the source of that ditty isn't my interest in physics or mathematics, but rather my interest in human cognition. Recent research indicates that both e and the Golden Ratio seem to play a role. Combine that with the rule of 7 +/- 2 (Miller, 1956) as well as Sir Roger Penrose's hypothesis that the mathematics of quasi-crystallization can be applied to human brain plasticity, and then ask yourself the question:

Is humankind distinct from nature, or just one small, however amazing, sliver of nature? If not distinct, then did nature apply one set of rules to us and another to the rest of nature? If no, then in principle, humankind and all of it's products, material and immaterial alike, from buildings to social networks, properly become (hard) scientific objects of study. And the rules that apply to the physical world, may also, to varying degrees of efficacy, be applied to the immaterial world and vice-versa.

This isn't a new idea. Durkheim, the Sociology equivalent of Albert Einstein in many respects, was saying the same thing, more or less, about a hundred years ago:

Man is not an empire within an empire
- Emile Durkheim (Elementary Forms of Religious Life)

And E.O .Wilson, evolutionary biologist and author of "Consilience" has been saying the same thing for years now to anyone who will listen.

Rather insanely, and more than a little baffling to me, I have been censored on more than one occasion by scientists (quite well educated and well meaning ones at that...) to whom such thoughts are heresy, a criminal offense tantamount to suggesting that the Earth is round (back in the days before we became "enlightened"). This, even as higher maths are being used to demonstrate the evolutionary basis of Cooperation...

Nice Guys Finish First
by David Brooks
May 16, 2011
The New York Times
http://www.nytimes.com/2011/05/17/opinion/17brooks.html

Oh, and the metaphor I employ?

E = PI^2

It's my evolutionary adaptation of e = mc^2. Just substitute "Power" for m and "Information" for c. As a hypothetical, exploratory construct, P maps to Power Laws (related to e) and I maps to optimal flow of information (related to phi). Where P or I equals 0, then E, effectively, equals 0 (there's actually another part of e = mc^2 most people aren't familiar with, which is why a photon has no mass, but does have energy...), leading to the following statement most activists are quite familiar with:

SILENCE = DEATH

- RF

RELATED PAPER
Period Concatenation Underlies Interactions between Gamma and Beta Rhythms in Neocortex
Roopun, Kramer et al.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2525927/

RECOMMENDED
"The Blank Slate: The Modern Denial of Human Nature " by Steven Pinker.
https://www.amazon.com/dp/0142003344/?tag=pfamazon01-20

A few other thinkers of interest: Carl Jung, Sigmund Freud, Daniel Dennett, Richard Dawkins, Albert Laszlo Barabasi, Clay Shirky, Duncan Watts...

 

[Raphie]

Tying together a few seemingly disparate concepts for you Jeremy...

Check out the Statistics version of the Pythagorean Theorem... Variance. A + B = C.

VAR(X) = E[X]^2 - E[X^2]

Also note the following symmetrical equation form: x^2 + 2xy + y^2 = z.

RELATED LINKS:

Variance
http://en.wikipedia.org/wiki/Variance

The Expectation Operator
http://arnoldkling.com/apstats/expect.html
E(X+Y)^2 = E(X^2 + Y^2 + 2XY)

Expected value
http://en.wikipedia.org/wiki/Expected_valueRF

PROBABILITY
--------------------------------------------------------------
x^2 + 2*(a*x) + a^2 = 1

Heads or tails...
.5^2 + 2*(.5*.5) + .5^2 = 1

Pareto's Law...
.8^2 + 2*(.8*.2) + .2^2 = 1

PALINDROMIC FORM
--------------------------------------------------------------
sqrt (10^2*7^2 + 10^1*2*(1*7) + 10^0*(1)^2) = 71
sqrt (10^0*7^2 + 10^1*2*(1*7) + 10^2*(1)^2) = 17
17 + 71 = 88 = 2*T_9 - 2

sqrt (10^0*9^2 + 10^1*2*(1*9) + 10^2*(1)^2) = 19 = sqrt (0081 + 180 + 100)
sqrt (10^2*9^2 + 10^1*2*(1*9) + 10^0*(2)^2) = 91 = sqrt (8100 + 180 + 001)
19 + 91 = 110 = 2*T_10

PALINDROMIC FORM w/ "Interference"
--------------------------------------------------------------
sqrt (10^0*11^2 + 10^1*2*(1*11) + 10^2*(1)^2) = 021 = sqrt (00121 + 220 + 100)
sqrt (10^2*11^2 + 10^1*2*(1*11) + 10^0*(1)^2) = 111 = sqrt (12100 + 220 + 001)
21 + 111 = 132 = 2*T_11

sqrt (10^0*13^2 + 10^1*2*(1*13) + 10^2*(1)^2) = 023
sqrt (10^2*13^2 + 10^1*2*(1*13) + 10^1*(1)^2) = 131
23 + 131 = 154 = 2*T_12 - 2

form: (square root of...) Parabolic Cyllinder + (square root of...) Parabolic Cyllinder

Sum of Central terms for 10,1 & 11,1
= 180 + 180 = 360 (--> Fundamental Domain of the Crystallographic Restriction Theorem)
= 220 + 220 = 440 (--> "Arbitrary" Reference frame for the Western Musical Scale = A-440 = 8*T_10 = 8*F_10 = 8*Ceiling [sqrt e^8])

(x + y)^2
--------------------------------------------------------------
(9^2 + 2*(1*9) + 1^2) = 100
(1^2 + 2*(9*1) + 9^2) = 100

etc...

THE INVENTOR OF THE PENDULUM CLOCK & EXPECTED VALUE
----------------------------------------------------------------------
Via Wikipedia...
Excerpt # 1
The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points. This problem is: how to divide the stakes in a fair way between two players who have to end their game before it's properly finished?

Excerpt # 2
Three years later, in 1657, a Dutch mathematician Christiaan Huygens, who had just visited Paris, published a treatise (see Huygens (1657)) "De ratiociniis in ludo aleæ" on probability theory. In this book he considered the problem of points and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens also extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players). In this sense this book can be seen as the first successful attempt of laying down the foundations of the theory of probability.
http://en.wikipedia.org/wiki/Expected_value

Christiaan Huygens
Christiaan Huygens... was a prominent Dutch mathematician, astronomer, physicist and horologist. His work included early telescopic studies elucidating the nature of the rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping, and studies of both optics and the centrifugal force.
http://en.wikipedia.org/wiki/Christiaan_Huygens

Huygens achieved note for his argument that light consists of waves,[1] now known as the Huygens–Fresnel principle, which two centuries later became instrumental in the understanding of wave-particle duality.

- RF