Prime Indices & the Divisors of (p'_n - 1): A Lattice-Related Question

[Raphie]

Prime Indices & the Divisors of p'_n - 1 such that Divisors & Indices are equivalent

for...
p'_n an element of {N} | d'(n) < or = to 2
This set of integers is equivalent to 1 Union the Prime Numbers aka "Primes at the beginning of the 20th Century"

where...
d'_n denotes the number of divisors of n < or = to n
Thus, by this definition d'(0) = 0 since 0 cannot divide 0 and all other N are greater than 0

Proposition:
n an element of {N} | d'(p'_n - 1) = n
n = 0,1,2,3,4,6
(and no other?)

The truth of this statement seems to me apparent, because the increase in the prime number indices far outpaces the growth of the divisors associated with them. For example, 5040 (7!) is the first integer with 60 divisors, and 5041 is the 675th prime. By comparison, 1081079 is the 84357th prime and is one less than the first integer with 256 divisors. By the time you get up to 11! = 39916800 (with a "mere" 540 divisors), you're at the 2428957th prime - 1. It seems logical to me that the number of divisors is just never going to "catch up" to the index number of the nth prime after n = 6.

So... what piece of key information am I missing here to be able to prove (or disprove) that which appears obvious?

TIA,
RaphieA Generalized Form of the Above:
n an element of {N} | d'(p'_n - 1) = n/(k!/|k-1|!)

e.g.
d'(p'_00 - 1) = d'(001 - 1) = d'(000) = 00/1 = 00; k = 0, n = 00
d'(p'_01 - 1) = d'(002 - 1) = d'(001) = 01/1 = 01; k = 1, n = 01
d'(p'_02 - 1) = d'(003 - 1) = d'(002) = 02/1 = 02; k = 1, n = 02
d'(p'_03 - 1) = d'(005 - 1) = d'(004) = 03/1 = 03; k = 1, n = 03
d'(p'_04 - 1) = d'(007 - 1) = d'(006) = 04/1 = 04; k = 1, n = 04
d'(p'_06 - 1) = d'(013 - 1) = d'(012) = 06/1 = 06; k = 1, n = 06
d'(p'_24 - 1) = d'(089 - 1) = d'(088) = 24/2 = 12; k = 2, n = 24 --> d'(n) is Monotonic (strictly increasing) to here
------------------------------------------------------------------------------
d'(p'_28 - 1) = d'(107 - 1) = d'(106) = 28/7 = 04; k = 7, n = 28
d'(p'_30 - 1) = d'(113 - 1) = d'(112) = 30/3 = 10; k = 3, n = 30
d'(p'_32 - 1) = d'(131 - 1) = d'(130) = 32/4 = 08; k = 4, n = 32
d'(p'_36 - 1) = d'(151 - 1) = d'(150) = 36/3 = 12; k = 3, n = 36
etc.

Then for k = 0 or 1...

n = 0, 1, 2, 3, 4, 6
d'(p'_n - 1) = 0, 1, 2, 3, 4, 6
p'_n = 1, 2, 3, 5, 7, 13

And a few observations re: 1,2,3,4,6 (special case of k = 1)
-------------------------------------------------------------------

A) 1,2,3,4,6 is the complete set of integers > 0 with a totient of 1 or 2

B) 1,2,3,4,6 is the complete set of the proper divisors of 12

C) Let a_n = 1,2,3,4,6, LK_n denote "Laminated Lattice Kissing Number" for n = (1-->4) Then...

((a_n * a_(n+1)) + 0) * (a_n^0 + 0)
= {02, 06, 012, 024}
= K_(a_n + 0)

((a_n * a_(n+1)) + 6) * (a_n^1 + 4)
= {40, 72, 126, 240}
= K_(a_n + 4)

This is the complete set of Laminated Lattice Kissing Numbers up to Dimension 8, and also the highest known Non-Lattice Kissing Numbers up to Dimension 8

TWO RELATED SEQUENCES
-------------------------------

n(n+ceiling(2^n/12))
0, 2, 6, 12, 24, 40, 72, 126, 240, 468, 960, 2002...
Sloane's A029929 http://www.research.att.com/~njas/sequences/A029929

Maximal kissing number of n-dimensional laminated lattice
0, 2, 6, 12, 24, 40, 72, 126, 240, 272, 336, 438, 648, 906...
Sloane's A002336 http://www.research.att.com/~njas/sequences/A002336

 

[Eynstone]

Proposition:
n an element of {N} | d'(p'_n - 1) = n
n = 0,1,2,3,4,6
(and no other?)

This follows from d'(p'_n - 1) <p'_n - 1< n for n>6.

 

[Raphie]

This follows from d'(p'_n - 1) <p'_n - 1< n for n>6.

Not sure what you are getting at Eynstone. Getting rid of the extra non-standard notational "gunk" so that p_n denotes nth prime and d(n) denotes the number of divisors of n:

d(p_n - 1) < p_n - 1< n for n>6

Let n = 8, then...
d(p_8 - 1) < p_8 - 1< 8 for n>6
d(19 - 1) < (19 - 1) < 8 and n>6

--> d(18) < 18 < 8
--> 6 < 18 < 8

... which is not accurate, obviously, since 18 > 8.

 

[CRGreathouse]

Your notation is odd (you write "d'(n)" for "d(n)", and "is an element of {N}" for "is an element of N", and don't specify the order of your p'), which makes it more work to follow than if you'd used standard notation.

As far as I can tell you're saying

Conjecture: $$d(p_n-1)=n$$ exactly when n is in {1, 2, 3, 4, 6}.​

or equivalently

Conjecture: $$d(p-1)=\pi(p)$$ for prime p exactly when p is in {2, 3, 5, 7, 13}.​

Of course you could modify these to work with 1 as well, but that wouldn't accomplish much.

This conjecture is easy to prove using the inequality $d(n)\le2\sqrt n.$

 

[Raphie]

Your notation is odd (you write "d'(n)" for "d(n)", and "is an element of {N}" for "is an element of N", and don't specify the order of your p'), which makes it more work to follow than if you'd used standard notation.

As far as I can tell you're saying

Conjecture: $$d(p_n-1)=n$$ exactly when n is in {1, 2, 3, 4, 6}.​

or equivalently

Conjecture: $$d(p-1)=\pi(p)$$ for prime p exactly when p is in {2, 3, 5, 7, 13}.​

Of course you could modify these to work with 1 as well, but that wouldn't accomplish much.

This conjecture is easy to prove using the inequality $d(n)\le2\sqrt n.$

First of all, CRGreathouse, thank you for the response, both regarding the proof and also the notation. You will find that, as someone who is self-taught, and has adopted an (experimental) observation- over proof- oriented approach to mathematics, I often use "odd" notation in order that I might describe such observations. But I am more than happy to try to bring my notation "into alignment" with standard notation whenever and wherever possible.

Secondly, re:

This conjecture is easy to prove using the inequality $d(n)\le2\sqrt n.$

Might you be able to refer me to any sources, online or offline where I might learn more about this inequality?

Thirdly, as for "not accomplishing much" by including the "trivial" case of 1, I will simply reply that I come across many relationships (descriptively speaking) by doing so; relationships I would not come across if I did not include 1.

In relation to this point, below the signoff is one small example I came across just yesterday demonstrating why I believe that including 1 can at times be "useful" insofar as it helps one to make observations that might otherwise be missed.

Lacking any accepted terminology, I call the below "Leech Lattice prime pairs," or, in other words, pairs of "primes" (including p'_0 = 1) that sum or subtract to 196560. I assume there to be an infinite number of such pairs, but this particular series is centered very close to "the origin," so to speak (196560 -11) and consists of a string of 19 consecutive prime pairs with additive or subtractive sum equal to 196560 (total unique values = 34).

I could show you a similar series in relation to (consecutive) Primes (including p'_0 = 1) that pair with another prime to sum or subtract to 90, including 1 + 89 --> (F_2 & F_11) --> (p'_0 & p'_24)Raphie

196560 = K_24 (Vertices of the Leech Lattice)
========================================
= p'_17689 + p'_32
-------------------------------------
= p'_17690 + p'_(5)^2
-------------------------------------
= p'_17691 + p'_28
= p'_17692 + p'_26
= p'_17693 + p'_23
= p'_17694 + p'_18
= p'_17695 + p'_17
= p'_17696 + p'_13
= p'_17697 + p'_12
= p'_17698 + p'_8
= p'_17699 + p'_7
-------------------------------------
= p'_17700 + p'_(5)^1
-------------------------------------
= p'_17701 - p'_0
= p'_17702 - p'_8
= p'_17703 - p'_9
= p'_17704 - p'_12
= p'_17705 - p'_16
= p'_17706 - p'_23
= p'_17707 - p'_25
= p'_17708 - p'_26
= p'_17709 - p'_27
-------------------------------------
= p'_17710 - p'_(5)^2
-------------------------------------
= p'_17711 - p'_31
========================================
See: Nth Prime Page (by Andrew Booker) http://primes.utm.edu/nthprime/index.php#nth

OBSERVATIONS

Re: 17700 (The "Centerpoint")
17700 = (8128(base 8))_(base 10)
17700 = F_11*L_11 - 11 = (89 * 199) - 11 = (p_(35 - 11 - 0) * p_(35 + 11 + 0)) - 11
for F_n & L_n denote, respectively, Fibonacci & Lucas Numbers; F_n * L_n = F_2n

and 8128 = 4th Perfect Number = T_127 = T_(p_31)
for T_n denotes Triangular Number (n^2 + n)/2; 31 and 127 are both Double Mersenne Primes (2^5 - 1 and 2^7 - 1)

8128 = ((4^2*L_4^2)/10^0 - (4^1*L_4^1)/10^-2)/4 = ((16*2207)/10^0 - (4*7)/10^-2)/4

Last term in this particular sequence
p'_(F_11*L_11) - p'_G_5
= p'_(F_(L_5)*L_(L_5)) - p'_(G_5)
= p'_17711 - p'_31
= 196687 - 127
= 196560
for G_n denotes Golden Scale Number (Fibonacci (2,5)) = the sum of F_(n-2) --> F_(n+2); 5 consecutive terms

"Distance" between prime indices (17711 - (2n + 1)); n = 1,2,3,4 and indices where paired primes are equivalent
Distance = 16,13,7 & 4 --> p_{26, 23, 12 & 8} --> p_{(5 + 21), (5 + 18), (5 + 7) & (5 + 3)}
Question: Possible Relationship with Cyclotomic Graphs of order 16, 13, 7, 4?
http://mathworld.wolfram.com/CyclotomicGraph.html

Also related...

Unique p' Divisors of 196560 = {1,2,3,5,7,13}
d'({1,2,3,5,7,13} - 1) = {0,1,2,3,4,6}
(1*2*3*5*7*13) * (0!/1!*1/0!*2!/1!*3!/2!*4!/3!*6!/5!)
= 2 * 196560
= K_1 * K_24
= (K_0 + K_0) + (K_24 + K_24)
= (K_phi(0) + K_phi(0)) + (K_phi(35) + K_phi(35))
= (K_phi(0) + K_phi(0)) + (K_phi(0 + 11 + 24) + K_phi(0 + 11 + 24))
for phi (0) = 0 according to Mathematica & 0, 11, 24 are terms associated with the Rydberg-Ritz Hydrogen Emission Spectrum

Related Thread: AN ALTERNATE VIEW OF THE PERIODIC TABLE
https://www.physicsforums.com/showthread.php?t=439315

 

[CRGreathouse]

But I am more than happy to try to bring my notation "into alignment" with standard notation whenever and wherever possible.

That was my purpose in pointing it out -- to help you better communicate with mathematicians (of all levels).

As for "not accomplishing much" by including the "trivial" case of 1, I will simply reply that I come across many relationships (descriptively speaking) by doing so; relationships I would not come across if I did not include 1.

It wasn't a criticism, just an explanation of why I dropped it in my 'translation' into standard notation.

Lacking any accepted terminology, I call the below "Leech Lattice prime pairs," or, in other words, pairs of "primes" (including 1) with interval = 196560. I assume there to be an infinite number of such pairs, but this particular series is centered very close to "the origin," so to speak. 196560 -11.

I really don't understand this 'definition' at all. Perhaps I'll understand when I read your examples. Does this mean pairs (p, p + 196560) where both are prime? ("prime constellations") Or perhaps such pairs of consecutive primes? Or something else entirely?

196560 = K_24 (Vertices of the Leech Lattice)
========================================
= p'_17689 + p'_32
-------------------------------------
= p'_17690 + p'_(5)^2
-------------------------------------
= p'_17691 + p'_28
= p'_17692 + p'_26
= p'_17693 + p'_23
= p'_17694 + p'_18
= p'_17695 + p'_17
= p'_17696 + p'_13

Ah. These seem to be pairs (p, q) with p + q = 196560. These are called Goldbach partitions (of 196560). There are 3663 such partitions.

Standard estimates exist for the approximate number of Goldbach partitions of a given number, if you're interested.

 

[Raphie]

That was my purpose in pointing it out -- to help you better communicate with mathematicians (of all levels).

It wasn't a criticism, just an explanation of why I dropped it in my 'translation' into standard notation.

Standard estimates exist for the approximate number of Goldbach partitions of a given number, if you're interested.

Make no mistake, I very much appreciate all of your feedback, CRGreathouse. Your notation is very clear and precise and it's great to have these examples of it that you have been offering.

[Edit]Although I am familiar with Goldbach's comet, I had never heard of "Goldbach partitions," or rather, I had forgotten the name, since I now recall that Calvin Clawson discusses them in his book "Mathematical Mysteries."In any case...[/Edit] Thank you for mentioning it. I'll have to see if I can find those estimates.

Also, I edited the definition you quoted. "Interval" wasn't the proper term I should have used since this only applies in the subtractive case, not the additive one. Here is how I rephrased it...

Lacking any accepted terminology, I call the below "Leech Lattice prime pairs," or, in other words, pairs of "primes" (including p'_0 = 1) that sum or subtract to 196560. I assume there to be an infinite number of such pairs, but this particular series is centered very close to "the origin," so to speak (196560 -11) and consists of a string of 19 consecutive prime pairs with additive or subtractive sum equal to 196560 (total unique values = 34).

Now, of course, I can call, at least the first half of the above sequence, by it's proper name (A subset of the...) "Goldbach Partitions of The Leech Lattice Vertices"Raphie

 

[Raphie]

By the way, CRGreathouse, it's funny you mention Goldbach because my original purpose in including 1 as the "zeroeth prime" was that I wanted to "see" integer sequences as both Goldbach and Gauss did in their time.

But here's a question for you:

What do you call it when you are also looking at (p-q) as well as (p+q), specifically in reference to consecutive primes that can subtractively or additively "pair" to a given integer?

e.g.

----------------------------------------------------------------------------
(5 + 85) or (- 5 + 95); 85 & 95 are both composite
----------------------------------------------------------------------------
7 + 83 = 90
11 + 79 = 90
-13 + 103 = 90
17 + 73 = 90
19 + 71 = 90
23 + 67 = 90
29 + 61 = 90
31 + 59 = 90
37 + 53 = 90
--------------------------------
-41 + 131 = 90
--------------------------------
43 + 47 = 90
47 + 43 = 90
53 + 37 = 90
59 + 31 = 90
61 + 29 = 90
67 + 23 = 90
71 + 19 = 90
73 + 17 = 90
79 + 11 = 90
----------------
83 + 07 = 90
89 + 01 = 90
----------------
97 - 07 = 90
----------------
101 - 11 = 90
103 - 13 = 90
107 - 17 = 90
109 - 19 = 90
113 - 23 = 90
127 - 37 = 90
131 - 41 = 90
137 - 47 = 90
--------------------------------
- 139 + 229 = 90
--------------------------------
149 - 59 = 90
- 151 + 241 = 90
157 - 67 = 90
163 - 73 = 90
- 167 + 257 = 90
173 - 83 = 90
179 - 89 = 90
- 181 + 271 = 90
191 - 101 = 90
193 - 103 = 90
197 - 107 = 90
199 - 109 = 90
----------------------------------------------------------------------------
(211 - 121) or (-211 + 301); 121 & 301 are both composite
----------------------------------------------------------------------------

Full Run = (20 + 1 + 22) = 43 Consecutive Primes (inclusive of p'_0)
START = 7; 7 = L_4
MIDDLE = 97 (- 7); 7 = L_4
END = 199; 199 = L_11
Range = p_4 --> p_46

Middle Run = (10 + 1 + 9) = 20 Consecutive "Primes" (inclusive of p'_0)
MIDDLE START = 43 (+47); 47 = L_8
MIDDLE 1 = 83 (+7); 7 = L_4
MIDDLE 2 = 89 (+1); 1 = L_1
MIDDLE END = 137 (-47); 47 = L_8
Range = p_14 --> p_33

 

[CRGreathouse]

What do you call it when you are also looking at (p-q) as well as (p+q), specifically in reference to consecutive primes that can subtractively or additively "pair" to a given integer?

No special name. They're no longer partitions, since you're allowing nonpositive parts, and no longer individually prime. You'd just say numbers that sum to n.

 

[Raphie]

No special name. They're no longer partitions, since you're allowing nonpositive parts, and no longer individually prime. You'd just say numbers that sum to n.

CRG,

While I hear you in regards to non-positive parts, on a philosophical note, it seems to me about as silly for modern mathematics to not allow for "negative partitions" as it would be to not allow convex or concave surfaces to be be called "surfaces", or as it once was to not allow negative or complex numbers (or even the number 1 as per the Greeks...) to be called "numbers."

But more practically speaking, I'm not getting what you mean by the phrase "no longer individually prime". With the exception of 1, all of the parts, meaning every p and every q of the example above are prime numbers, unless, that is, I made a "mind-typo" somewhere along the way, which I've been known to do. That seems to me a very different manner of statement than simply stating 121 - 31 = 90.


Raphie

 

[CRGreathouse]

While I hear you in regards to non-positive parts, on a philosophical note, it seems to me about as silly for modern mathematics to not allow for "negative partitions"

Remember, we're talking about a name here -- partition -- not about what is considered reasonable.

If you allow nonpositive parts, there are infinitely many 'partitions', which means you give up things like counting them! This is, of course, the reason that partitions have that special name.

But more practically speaking, I'm not getting what you mean by the phrase "no longer individually prime".

Well, I guess this comes from your definition of prime. I don't consider -5 prime.

Generally this depends on whether you come from an algebra* background (where -5 is prime) or a number theory or combinatorics background (where -5 is not prime).


* Abstract algebra, that is, not the high school stuff.

 

[Raphie]

OBSERVATION

While I've known for some time that 1,2,3,4,6 are the only allowable n-fold rotational symmetries under the crystallographic restriction theorem, I only just now happened to come across the below mathematical formula that can be used to prove the theorem, surprising to me in it's simplicity...

|2*cos (2*pi/n)| is in N

Solutions: 1,2,3,4,6

e.g. |2*cos (2*pi/10)| = phi, not an integer (but related to quasicrystallization; see: How to Make a Quasicrystal http://www.physics.emory.edu/~weeks/software/exquasi.html)

Source: Crystallographic restriction theorem: Short mathematical proof (via Wikipedia)
http://en.wikipedia.org/wiki/Crystallographic_restriction_theorem#Short_mathematical_proof

- RFPosted in regards to the following identity, which is the topic of this thread:

n | d(p_n - 1) = n --> {1,2,3,4,6}

for p = 2,3,5,7,13 ("Frampton-Kephart" Primes)
see: Mersenne Primes, Polygonal Anomalies and String Theory Classification http://arxiv.org/abs/hep-th/9904212