perfect numbers

[AntonioLao]

According to mathforum.org the list below are the first 15 perfect numbers. A perfect number is defined as the sum of its factors less than the number is equal to the number. For example, the factors of 6 are 1, 2, and 3. The sum 1+2+3=6. The number 28 has factors 1, 2, 4, 7, and 14 their sum 1+2+4+7+14=28. One obvious property of perfect number is that it is always an even number. If infinity is an even number then it is the largest perfect number. On the other hand, the prime factors of 6 are 2 and 3. The prime factors of 28 are 2, 2, and 7. The product of prime factors must always equal the number: 2x3=6 and 2x2x7=28.
1. 6,
2. 28,
3. 496,
4. 8128,
5. 33550336,
6. 8589869056,
7. 137438691328,
8. 2305843008139952128,
9. 2658455991569831744654692615953842176,
10. 19156194260823610729479337808430363813099732154816 9216,
11. 13164036458569648337239753460458722910223472318386 943117783728128,
12. 14474011154664524427946373126085988481573677491474 835889066354349131199152128,
13. 23562723457267347065789548996709904988477547858392 60071014302759750633728317862223973036553960260056 13602555664625032701750528925780432155433824984287 77152427010394496918664028644534128033831439790236 83862403317143592235664321970310172071316352748729 87474006478019395871659364010874193756490579185494 92160 555646976,
14. 14105378370671206906320795808606318988148674351471 56678388386759999548677426523801141041933290376902 51561950568709829327164087724366370087116731268159 31365248745065243980587729620729744672329516665822 88469268077866528701889208678794514783645693139220 60370695064736073572378695176473055266826253284886 38371507297432446383530005313842946029657514336806 5570759537328128,
15. 54162526284365847412654465374391316140856490539031 69578460392081838720699415853485919899992105671992 19190573900802636461592800138276054397462627889030 57303445505827028395139475207769044924431494861729 43511312628083790493046274068171796046586734872099 25721905694655452996299198234310310926242444635477 89635441481391719816441605586788092147886677321398 75666162471455172696430221755428178425481731961195 16598555535739377889234051462223245067159791937573 72820860878214322052227584537552897476256179395176 62442631448031344693508520365758479824753602117288 04037830486028736212593137899949003366739415037472 24966984028240806042108690776703952592318946662736 15212775603535764707952250173858305171028603021234 89664785136394992890497329214510750597991145622151 9899345764984291328

[SteveA]

As you saw elsewhere, an interesting association here with 6 is that this can be reorganized into a 1 dimension string or layered in orthogonal directions to construct multidimensional volumes - if you had to serialize and a space to squeeze it through a quantum hole, then this could come in quite handy. For example with 6 being equal to 1+2+3 or 1*2*3, we could geometrically show this in a linear fashion using elements A, B and C to represent these 3 components:

ABBCCC (linear representation)

B-AAA
B-AAA
..|||
..CCC

In this case the element A is replicated in dimensions of width B and C to create a volume of 6 spaces as well and we could transform these 6 units into being either linear or a volume of orthogonal dimensions.

Imagine having all of these 3 components be cyclic rings rotating synchronously with periods, 1, 2 and 3. We could describe this similar to counting parallel incrementing digits in different bases (imagine independent photons orbiting within some closed number of quantum units and observing the state of the system):

Base 1:0000000000000...
Base 2:0101010101010...
Base 3:0120120120120...

The period of repetition is 6, which is the same quantity/volume as the space in which these motions occur - both the real and virtual volumes can have a 1 to 1 mapping constructed between them. To compute the period in general, we have to remove common divisors between these wavelengths.

In the case of 28 we have:

28=1+2+4+7+14

If we similarly split these into independent bases of representation, the repetitions are (I'll use an alphanumeric representation):

Base 1:(0)0000000000000000000000000000000000000000000000000000000(0)000
Base 2:(0)1010101010101010101010101010101010101010101010101010101(0)101
Base 4:(0)1230123012301230123012301230123012301230123012301230123(0)123
Base 7:(0)1234560123456012345601234560123456012345601234560123456(0)123
Base14:(0)123456789ABCD0123456789ABCD0123456789ABCD0123456789ABCD(0)123

The period of repetition is 56 (Iron 56 perhaps?) As you pointed out this is the most stable nuclear atom and the normal end result of fusion). If we look at 6, that's carbon and a rather key element in life.

As I've commented elsewhere I think observations should be able to be broken down into sequences of information in a single dimension of time. This correlation. Notice that if we began with 56 and constructed cyclic loops of all its factors, these would have a multidimensional structure with a volume of 56 as well and a 1 to 1 transformation between a multidimensional object and a single loop in 1 dimension could exist compatibly ... wow, there's something to that. Imagine, along the lines of string theory that we have a single closed loop orbital at a constant velocity. If this loop was 56 quantum units long, it would repeat every 56 quantum units of time. If we then proceeded to "twist" this loop so that it intersected itself and similar to generating harmonics with a string by holding stationary various segments of it, this newly "tangled" version of the same thread could have a virtual period with an equivalent spectrum that have a period of repetion equal to the initial length of loop.

That idea fits in very well with some of my other comments on the "Theory of Life, Consciousness and Computation" thread.

Actually, the same technique can be used with any number, though the uniqueness of perfect numbers is their potential to have both a serial or multidimensional representation as transformation of a single conserved volume (in a sense, that transformation could be seen as informationally creative in the sense that a conservation exists as well as the ability to append an additional context to it - in this case it would be binary, but there might be ways of having multiple possible transformations and generate a spectrum of relatively prime transformations).

My assumptions regarding chemical reactivity is to take, for example a loop length of 3. If we looked for any synchronized characteristics over time between this loop length and that of the 56 period loop, there is no correlation (the greatest common divisor between these is 1, or they share no common factors) and observing ones gives no information regarding the state of the other, hence they appear "neutral" to each other.

On the other hand, if we compared the length 3 loop to that of the length 6 loop we could find that knowing the state of the length 6 loop can always predict what state the length 3 loop is in and that the length 3 loop can always determine 1 of 2 possible positions for the length 6 loop. This gives us a structure similar to a 2:1 ratio in an atomic orbital. We also have 2 relatively prime numbers involved as 2 and 3, which allows information to exist in two "spectral" forms (we can't convert between these perfectly and that can be seen as the ratio of information being an irrational number log(3)/log(2)~=1.585 or we could similarly say that 2^n != 2^m with n or m greater than 0)

An interesting "qualitative" feature to such a view is that, in effect, all of these loops can exist entirely independently and interactions are observer dependent ... (free will, subjective divergences etc.?).

[SteveA]

Oops, it looks like I made a mistake regarding some of my comments about how a quantity of 56 could appear synchronized (I guess it would have a period of 112 instead), though we could construct a subset of factors that works ... and it might be this ability that gives some freedom in chemical and atomic structures. Anyway, that's another area to explore.

It's ironic that in terms of nuclear processes iron is least providing of energy, yet in terms of our metabolism iron is seen as a key receptor of energy in the body.

[AntonioLao]

In this sense, Nature is smarter than us since the snow crystal is hexagonal and the earliest living fossil ( the honey bees) also build their honeycomb as the tessellation of hexagons. In three dimension, the 6 vertices polyhedron is an octahedron and I use it to represent the 6 spacetime charges of the up quark. The properties of octahedron can be found in this link:
http://mathworld.wolfram.com/Octahedron.html and http://en.wikipedia.org/wiki/Octahedron

[AntonioLao]

Steve, the following excerpt from one of my earlier threads shows the perfectability of 6 in the imaginary complex domain as well:


The prime factors of the number 1995 are 3, 5, 7, and 19. However, the prime ideal factors of 3 are (1+√2i)(1-√2i). The prime ideals of 5 are (1+2i)(1-2i) or (2+i)(2-i). That of 7 are (1+√6i)(1-√6i) or (2+√3i)(2-√3i), and that of 19 are (1+√18i)(1-√18i) or (2+√15i)(2-√15i) or (3+√10i)(3-√10i) or (4+√3i)(4-√3i). On the other hand, mathematicians define perfect numbers as numbers whose factors sum are equal to the number themselves, for example, the factors of 6 are 1, 2, 3, and their sum is also 6. Another perfect number is 28. Its factors are 1, 2, 4, 7, and 14 whose sum is also 28. The prime ideal factors of 6 are (1+√5i)(1-√5i) or (2+√2i)(2-√2i) whose sum is also 6. The ideal factors of 28 are (1+√26i)(1-√26i) or (2+√24i)(2-√24i) or (3+√19i)(3-√19i) or (4+√12i)(4-√12i) or (5+√3i)(5-√3i) whose sum is 30, since 30≠28, 28 is not a perfect number in the complex domain. Therefore, taking both real and complex ideal prime factors only the real number 6 could really be considered as a truly perfect number. Coincidentally, the square ratio of 6x6 square singular Hadamard matrices for proton to electron is the product of 2 and the 8th power of 6 whose square root is approximately the mass ratio of proton to electron.

[SteveA]

Your comments here regarding complex primes (gaussian primes?) is interesting and reminds me of an idea I was experimenting with concerning spiraling vectors in 3 dimensions.

Imagine an aircraft flying at a constant velocity but with 2 constant rotational/angular velocities, such as a fixed rate of roll and yaw. Some combinations of these can lead to spiral trajectories away from an origin, others can return to an origin (with an alignment that is potentially not the same, in which case various loops can reintersect the origin in complex ways).

When I plotted out the "stability" of these in terms of which values for these angular rotations tended to remain near the origin, this 2 dimensional stability graph appeared quite similar to the chart of atomic stabilities (including a couple "islands" of stability off the tip of the "peninsula"). So we could figuratively equate a proton as a unit of rotation in one dimension and a neutron as a rotation in the other dimension. It may be that only one of these rotations need be quantized and the stable solutions in the other dimension of rotation would then potentially inherit such quantized characteristics indirectly (potentially mass could arise from irrational ratios involved between these over time? Just speculation)



This would appear to resemble extending the concept of complex primes into a 3 dimension realm, though I used real numbers and that's not ideal, IMO.

Just some more thoughts. Thanks for posting your ideas.

[AntonioLao]

I am wondering what would be the graph of the two orthogonal rotations of a helicopter with its additional ability to perform vertical takeoff?

[SteveA]

I am wondering what would be the graph of the two orthogonal rotations of a helicopter with its additional ability to perform vertical takeoff?

Well it would probably look like a curved 2-D surface but with a highly fractal/chaotic foam/form. It's an interesting idea to try rendering some of it.

Wouldn't it be nice if we could e-mail 3 dimensional objects? (Maybe make it solid and transparent but use internal color coding to indicate various features regarding the form of stability with a set of parameters ... for example, though a trajectory might return to an origin, it could reintersect the origin at a different orientation and take multiple "orbits" to return to an approximation of the origin orientation and we could have color coding similar to describing continued fractions http://mathworld.wolfram.com/ContinuedFraction.html and this could be like describing the wavelengths of nested orbital periods of a system approximating that period ... we could then compute approximate common divisors and potentially compute equivalent electromagnetic properties of the material relative to other materials arising from the appearance of synchronizations in time of these orbits relative to each other within such a material ... which then gives us another incredible level of complexity, that I've made some comments about before on my other thread. Rather amazing what combining a few simple ideas together could derive. My guess is that something like this is actually what's occuring in physics, though a factor to consider is space itself and much of the complexity might actually be derived not from the material of an object itself but its position within space)

[AntonioLao]

Advanced computer technology has already made all these possible. However, too much of these new 3D knowledge is used in virtual reality in computer games. The University of Texas in Dallas offers a course for this 3D game virtual reality. I'm hoping to write a proposal to get a funding for a 3D simulation using virtual reality to model a quantum theory of spacetime. The units of this virtual spacetime would be the H-pluses and the H-minuses with their intrinsic rotations with variable programs of length of radius, spatial and temporal frequency with a good pseudo random number generator. This might be similar to John Conway's algorithm for artificial life which I've written a simple working program using BASIC language in 1985.

[SteveA]

I LOVE BASIC I've coded in different languages, but BASIC is still my favorite. Visual Basic is good, but expensive.

The one I currently use that has a nice editor and is almost as fast as C code is FreeBASIC. There's an editor called FbEdit that works great with it and that's almost exclusive what I've been using to 'prototype' up ideas (I've been writing up some real time audio synthesis algorithms as well with it and those graphic images were done with it as well).

Here's a bundle of both FbEdit and FreeBASIC: http://fbedit.freebasic.net/viewtopi...da370139f2230c

That bundle could be dated though and a new version of FreeBASIC is probably available, but this package should have the editor and compiler working together well (at least I believe this is the version I've been using)

If you want some samples of code to experiment with, I can post some to mess around with.