Cosmology

[desiresjab]

I keep getting those "you are banned" messages when trying to post as it looks like you do as well.

If our subjectivity is not a game, then neither is mathematics. I would also elevate poetry to being something more than a game.

However, what comes out of our subjectivity is partial. We cannot take it too literally.

Yeah. I consider normal language much more complex than mathematics. Programming Big Blue to beat Kasparov in chess was fantastic, but the far more formidable task was programming it to beat Ken Jennings at jeopardy. The procedures of chess are rather mathematical at heart, jeopardy is not. I believe Big Blue was not allowed to read the questions--for that would have happened instantaneously, but had to hear them and understand them. Big Blue had to understand all the puns and allusions in the typical jeopardy question. This is much closer to understanding poetry than it is math.

Connotation and suggestion is so complex. The same images will not form in our minds as we read Shakespeare, the same thoughts. Words do not have equal signs between them, even synonyms do not. Every word is different from every other. The same word will have different connotations in a different setting. This not true of the number 6.

[YesNo]

The number 6 can mean all kinds of things to us. It can be a composite number. It can be one-third of the "beast" 666. It can suggest a six-pack of whatever we want at the moment.

[Dreamwoven]

I keep getting those "you are banned" messages when trying to post as it looks like you do as well.

This is weird, that both of you have got such banned messages

[YesNo]

Although I probably deserve it for all my hell-bent sins, it seems that we get them when we use special characters to format math concepts in a post and the software thinks we are using a browser that does not allow ads to display.

[Dreamwoven]

What is a special character? Like Chinese? It would be nice to be rid of ads, if that is possible.

[YesNo]

Like the symbols above the numbers on the keyboard. I figured it is best not to touch them at least when there are numbers next to them.

As far as ads go, I usually don't mind them. I have been known to click on one or two. They just have to display rapidly.

[desiresjab]

Like the symbols above the numbers on the keyboard. I figured it is best not to touch them at least when there are numbers next to them.

As far as ads go, I usually don't mind them. I have been known to click on one or two. They just have to display rapidly.

I am pretty sure that is not it. The site was recently restructing some stuff.

[desiresjab]

Yes/No, even now I ponder quadratic reciprocity everyday. I love the concreteness of it compared to abstact philosophical talk.

Like I said before, I am dumb, so it takes me a long time to see things.

But by putting a little bit next to a little bit, I have finally seen what I wanted to see, not more than five minutes ago for the first time.

No abstract algebra or group theory needed, just a minute inspection of the details of Eisenstein's proof.

One can see and understand almost all the details of Eisenstein's proof without understanding why it proves QR.

I had already figured out that the dimensions of the rectangle represented the scale of the relative sizes of the moduli working aginst each other in QR. What I had not put into words was that this representation of scale is only activated by the diagonal. Hold that thought.

* * * * *

The other thing is a clear concept of just what the multiplication

[(p-1)/2] [(q-1/2)] stands for. What does it stand for? First, each is the number of respective quadratic residues of the primes individually.

This multiplication stands for any one of 5 things combined with anyone of 3 things. In other words, it counts how many ways the number of quadratic residues of each prime can be combined with one another, fifteen ways, in this case.

Each combination has its chance. Each combination is a lattice point. The diagonal slices WAXY one more time, dividing the number of lattice points a final time. If it is slicing through an odd number of lattice points, triangles WAY and YAX are forced to different parities. This only happens if they are both 4n+3 primes.

The diagonal expresses the gear size of each prime. Set with the bigger prime as width rather than height, I can get many more lattice points in WAY than YAX with a large enough size discrepancy between primes. I am pretty sure of that conjecture. It is the polar opposite of my other conjecture.

Yes, p and q are the individual gears, but the diagonal is them meshed together.

[desiresjab]

We start with a p by q rectangle. We fill in the lattice points. We divide the rectangle by two vertically, and divide it by two again horizontally.

Only now do we divide it by two once more with the diagonal, allowing the diagonal to be "last agent," as you might prefer.

With the construction of the rectangle, the gears sizes are set. With the construction of the diagonal, the gears are meshed together and running.

Each gear is a period, a modular cycle of remainders. When you combine two periods, you get a larger period, like a period of 77 for pq, before everything is back to where it started. The original marks on the two gears will again be aligned vertically with a stationary reference point.

The upper triangle WAY can "hog" lattice points, because the extreme "lean" of the diagonal forces lattice points into the upper triangle WAY in the left hand lower corner of the rectangle, but the best the lower triangle YAX can ever do is break even.

[YesNo]

No abstract algebra or group theory needed, just a minute inspection of the details of Eisenstein's proof.

I agree that a better understanding should not need those tools. They help to generalize and perhaps prove results.

This multiplication stands for any one of 5 things combined with anyone of 3 things. In other words, it counts how many ways the number of quadratic residues of each prime can be combined with one another, fifteen ways, in this case.

Do you have an example? I don't follow the 5 and 3 things.

Each combination has its chance. Each combination is a lattice point. The diagonal slices WAXY one more time, dividing the number of lattice points a final time. If it is slicing through an odd number of lattice points, triangles WAY and YAX are forced to different parities. This only happens if they are both 4n+3 primes.

I agree. You will only get an odd number if both primes have remainder 3 modulo 4.

The diagonal expresses the gear size of each prime. Set with the bigger prime as width rather than height, I can get many more lattice points in WAY than YAX with a large enough size discrepancy between primes. I am pretty sure of that conjecture. It is the polar opposite of my other conjecture.

Is there an upper bound on this discrepancy?

Yes, p and q are the individual gears, but the diagonal is them meshed together.

"Gears" is a nice metaphor. I had not thought of it like that before.

[desiresjab]

I agree that a better understanding should not need those tools. They help to generalize and perhaps prove results.



Do you have an example? I don't follow the 5 and 3 things.



I agree. You will only get an odd number if both primes have remainder 3 modulo 4.



Is there an upper bound on this discrepancy?



"Gears" is a nice metaphor. I had not thought of it like that before.

5 and 3 are (p-1)/2 and (q-1)/2 when p=7 and q=11. The simple multiplication is counting the ways three objects can combine with five objects in pairs. There are fifteen different pairs representing how p can pair with q and vice versa. At this point the ground level mechanics are gone and we are looking for something else. We only need to keep our tether line connected to where we started from so we can remember where we are.

[YesNo]

I see. These numbers will change depending on the primes involved.

[desiresjab]

Duh, I must be slow. I have to admit, I either forgot or never realized that a simple multiplication represents how many ways the objects from two sets can be paired.

In QR I think it is important that (p-1)/2(q-1)/2 represents that, not just some normal product as we usually think of a multiplication. Basic multiplications are combinatorial, if you enlarge your viewpoint slightly. It gives something deeper to explore. If I can reverse map each of the fifteen lattice points... another revelation might be near. Ahem! A mirage is likely, too.

[desiresjab]

At least 227 proofs of QR are known. Even the few I know of use a staggering array of techniques and math. There are proofs emanting from:

1 Modular Arithemtic
2 The Pythagorean Theorem
3 Abstract Algebra
4 Group Theory
5 Geometry
6 Combinatorics
7 Trigonometry
8 The Binomial Theorem
9 Class Field Theory
10 Calculus (real anaysis)
11 Calculus (complex analysis)
12 Euclidian Algorithm
13 The Chinese Remainder Theorem
14 Vectors

Additional fields or functions that I suspect proofs emanate from:

1 Euler's Totient Function
2 Game Theory
3 The Divisor Function
4 Eliptic curves
5 Modular Functions
6 Discrete Logarithms
7 Primitive Roots
8 Fermat's Little Theorem
9 Statistics

Each of the fields probably has produced numerous proofs with slightly different twists. QR is so centrally connected, as I keep mentioning, or these diverse fields would not all have relations with it.

[desiresjab]

Of course each lattice point only represents any old pair of quadratic residues (from anything that is said in the proof). My experimental idea is to replace each lattice point in WAXY with a specific pair of residues There may be a revealing way of matching each particular lattice point to every specific residue pair. How to match them is an intriguing question, which I am hoping will later become obvious, because that would mean there is a superior way of mapping point to pair. I think I will carry out the number crunching. More later.