Cosmology

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Originally Posted by YesNo

One of our differences is that I don't view mathematics as more than a game. You seem to think there is more to it.

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You will call it a game, come tsunamis or asteroid hits, I know that. In that case I would at least like to hear form you that it is the game we found here, whose laws we did not invent but only formulated in more detail from what is abstractly necessary.

Which is it, my dear Yes/No? Did we wholly invent this game of mathematics, like we did chess, or did we find ineluctable rather than arbitrary rules were necessary even to do something as basic as counting properly in the universe we found ourselves in? You should be able to answer that without too much sophism about rules of a game.

1 The laws of Math are purely our inventions, just another game?

2 The laws of math are such that they must be as they are. We can discover and develop new byways in them but not invent the mechanics those laws rely upon.

Take a crack at it.

Let's consider the claim that "two follows one". I agree with you that that statement is true in every universe since none of those universes care one way or the other about it. Similarly, the game Tic-Tac-Toe is true in any universe. A limerick I write is a limerick in any universe.

To get more specific, What does it mean to say that two follows one? If we are in the integers it means that there is a binary relationship, in this case a strict total ordering, containing the pair (2,1) requiring that 2 > 1.

We could easily switch that relationship around and say that "one follows two" by changing the binary order relationship to include (1,2) and not (2,1). In general if a > b then we make b > a. That is also true in every universe, because the existence of any of those universes is not dependent upon how we define that binary order relationship.

We can go further. Consider the finite field of two equivalence classes labelled 0 and 1. 0 contains all the integers that are even and 1 contains all the integers that are odd. In that field the statement "two, as an equivalence class, does not exist" is true in all universes.

None of those universes care how we define the rules of the game we are playing at the moment.

The answer to 1 is that mathematics is a lot like a story or poem. It objectifies what we subjectively understand. That objectification I can call a "game" although it could be a "story" or a "limerick".

The answer to 2 is similar. When I write a limerick it conforms to a certain pattern or it is not a limerick. When I construct a mathematical structure, what I derive from it follows logical patterns or it is not a mathematical structure.

The missing question is does the universe behave like any of these mathematical structures (or stories or poems or physical theories) that we might happen to create? We hope it does, but we are led into errors when we take these objective artifacts of our subjectivity too literally. In spite of what some people like to believe, we cannot completely dump our subjectivity into something objective.

There is another problem. When we do not have physical evidence for something we sometimes rely on these objectified, mathematical structures to guide us. This creates a blind spot. For example, is time in reality a mathematical continuum of infinitesimal points? We cannot physically verify infinitesimals given quantum limitations to the discrete. Furthermore, the assumption that time is a mathematical continuum leads to paradoxes in physical reality, notably Zeno's, even though the paradoxes might have been resolved mathematically.

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Originally Posted by YesNo

Let's consider the claim that "two follows one". I agree with you that that statement is true in every universe since none of those universes care one way or the other about it. Similarly, the game Tic-Tac-Toe is true in any universe. A limerick I write is a limerick in any universe.

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Tic-Tac-Toe is true in any universe, but it is not much of a tool for exploring universes. Its laws, too, were always possible in any universe, but it is not fundamental to any of them

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Originally Posted by YesNo

To get more specific, What does it mean to say that two follows one? If we are in the integers it means that there is a binary relationship, in this case a strict total ordering, containing the pair (2,1) requiring that 2 > 1.

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This argument does not describe a universe, friend, nor does it have anything to do with the cardianl successor of 1. You ask what it means. It means two is the cardinal successor of 1. Your argument is totally irrelevant, a true red herring.

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Originally Posted by YesNo

We could easily switch that relationship around and say that "one follows two" by changing the binary order relationship to include (1,2) and not (2,1).

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And you think this has anything to do with cardinal successors? This is as relevant as looking at 18 and claiming 8 must be the successor of 1.

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Originally Posted by YesNo

In general if a > b then we make b > a. That is also true in every universe, because the existence of any of those universes is not dependent upon how we define that binary order relationship.

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Again, sir, we are not talking about binary relationships. Notice you are not counting.

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Originally Posted by YesNo

We can go further. Consider the finite field of two equivalence classes labelled 0 and 1. 0 contains all the integers that are even and 1 contains all the integers that are odd. In that field the statement "two, as an equivalence class, does not exist" is true in all universes.

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Successors, sir, fundamental counting--that's where we are. You are not counting in this example, you are blinking two values as in QR.

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Originally Posted by YesNo

None of those universes care how we define the rules of the game we are playing at the moment.

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What makes you think so? They care a lot, if you want to put it that way. They care enough that they will not let you change any laws, you have to use the ones that preceded your arrival. The universe does not care about Tic-Tac-Toe because Tic-Tac-Toe is not integral to it.

A man made up Tic-Tac-Toe. Did a man make up the mechanics that causes 4n+1 numbers to behave differently than 4n+3 numbers in QR? A man saw that they behaved that way, a man did not make up that they behaved that way. They were behaving that way always, before Euler conjectured it and before Gauss proved it. A man did not make up the fact that numbers will behave this way. there is no other way possible for them to behave. Ineluctable: can be no other way.[/QUOTE]

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Originally Posted by YesNo

The answer to 1 is that mathematics is a lot like a story or poem. It objectifies what we subjectively understand. That objectification I can call a "game" although it could be a "story" or a "limerick".

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Before, you said mathematics was a stroy, now you say it is a lot like a story or a poem. Make up your mind.

You are really astray and I do not know if I can help you. You are cruising at a very high level of abstraction making judgements on the lowest level in the universe. Because a mailbox and a car are both red, at a particular level of abstraction they are alike. Because a peach and a baby are both soft, at a certain level of abstraction they are the same. As you descend to lower levels the differences betwen babies and peaches becomes clear. They are only superficially alike. You see there are a lot more differences than there are similarities between the two.

You are making high level abstractions to point out superficial commonalities.

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Originally Posted by YesNo

The answer to 2 is similar. When I write a limerick it conforms to a certain pattern or it is not a limerick. When I construct a mathematical structure, what I derive from it follows logical patterns or it is not a mathematical structure.

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High level of abstraction to point out superficial similarity.

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Originally Posted by YesNo

The missing question is does the universe behave like any of these mathematical structures (or stories or poems or physical theories) that we might happen to create? We hope it does...

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Which part of the universe are you talking about? Again, youapparently only mean cosmological or quantum scales. Here at mammal-scale, math works just fine to capture order and help us make better decision by the millions everyday.

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Originally Posted by YesNo

There is another problem. When we do not have physical evidence for something we sometimes rely on these objectified, mathematical structures to guide us. This creates a blind spot. For example, is time in reality a mathematical continuum of infinitesimal points? We cannot physically verify infinitesimals given quantum limitations to the discrete. Furthermore, the assumption that time is a mathematical continuum leads to paradoxes in physical reality, notably Zeno's, even though the paradoxes might have been resolved mathematically.

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You are like a madman that only knows certain things to repeat. There is no blind spot created by math. Infinitesimals cannot be verified. What are you going to do, find a smallest one?

Most of the mechanics of the universe were a blind spot to early man. Math reduces and illuminates them, it does not create them.Your problem is you are disappointed the lighting is not perfect. You are mad because when the light of mathematics is shined on the universe it does not show everything with 100% resolution. We have to know more laws than we do now. When those laws of numbers are found, some of them may apply to your obsession. Until then, hope for a great mathematician to arise, because the solution if it comes will be in mathematics, it will not come from people with prayer books. The people with prayer books are not even searching for the same kind of answer. I would consider it a miracle if people praying were ever of instrumental use in science.

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Originally Posted by desiresjab

This argument does not describe a universe, friend, nor does it have anything to do with the cardianl successor of 1. You ask what it means. It means two is the cardinal successor of 1.

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Where is the number 1 in the universe?

It is not anyplace.

I see. It is a game we created. My cat doesn't play the game (to my knowledge). I don't even think my computer really knows what the number 1 is.

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Originally Posted by YesNo

I see. It is a game we created. My cat doesn't play the game (to my knowledge). I don't even think my computer really knows what the number 1 is.

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Am I still banned?

I don't know if you got the question out just right. But since it is the big question in all of this, I have to address it.

If we start with 1, it would not hurt to say what 1 is, right? I think that is what you meant.

Words like "unity" or "singularity" only stall the answer. They are not what a person who asks the question means to get at. Those two words already contain the notion of oneness, so we cannot use them to define oneness, can we now? I mean really.

Everything else is defined neatly in terms of 1. But what is 1? That is it, isn't it? Here is where it gets thorny philosohically.

Everything after 1 is mathematically defined, but is 1 mathematically defined?

Even though there is a mathematical operation 0+1=1, that would be getting ahead of ourselves.

The idea of 1 seems to be philosophical. Translated to counting, it can only be existence itself, as opposed to no existence. 1 and 0 are existence and non existence, mathematically. Ah! but that is one level of abstraction up from the abstract concept of existence itself, isn't it? I just learned something.

Further, the reason mathematicians did not start at 0 to define numbers is that they would then would have to explain existence itself when 1 appeared from nowhere.

1 is the assumption of existence, which is the foundational assumption of mathematics, maybe of philosophy too. Once we do that, everything follows nicely, in math at least, and we can even go back at that point and say, yes, 0+1=1.

Unless you must call the assumption of existence a game, where is the game? We conceive of nothingness, observe we exist, and further observe there is a proliferation of things that exist. If you want to enumerate things that exist accurately you must use numbers to count them. Each number is the successor of the number in front of it, exceeding it by exactly one, the unit already defined and brought out of nothingness to represent existence. One is one of anything—a car, a house, a dumpling, an idea.

We know we exist. I myself believe we are not what we think we are by a long shot, but in one way or another we and other things do exist. We have every right to assume that we exist and that other things exist as well.

We invented symbols for existence and non existence, 1 and 0. Man was well along in the civilization process before the idea of 0 came and stuck.

The truth is, historically mankind started with one, when it came to counting, since it was a reasonable assumption that things existed. It was a long time later when they got around to explaining how zero worked in every situation. Very little in math ever came easy after counting. Counting was pretty natural, in the sense that enumeration could easily be verified with the senses, but counting still took a long while to develop anyway.

I do not see counting as a game, nor the assumption of existence as making a rule. I do not see keeping track of enmueration a game.

Math can be used like a game, one can think creatively and inventively with it, but the fundamental propositions of mathematics are no game, they came right out of counting what was already there using number successors before anyone ever bothered with the concept of number successors.

It is OK to view nature as your opponent to be mastered, but you would not want to mistake any model for the actual, even the game model for mathemathics, right?

Math can be fun like a game. I want it to be. It is to many.

Even non commutivity in multiplication came about to satisfy multiplication of matrices and groups, not as an arbitrary experiment with operations. Matrices are shadowed by much in nature and our daily lives. Matrices are a brand of mathematics that does not count things, but the magnitudes of the numbers in the columns and rows of the matrix are interpreted exactly as in counting. 26 is still 26.

It is all right to approach mathematics like a game when you do math research. It should be fun. Satisfying your curiosity is a fun thing.

In order for matrices to reflect the way certain things happen, one operation (multiplication) had to be tweaked. Galois was following nature, maybe looking for it.

Since the whole dispute in my mind now comes down to the number 1 and how it got there, I can more easily make my points.

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.

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Originally Posted by YesNo

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.

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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.

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.

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Originally Posted by YesNo

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

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

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.

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

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.

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Originally Posted by 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.

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

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.

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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.

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.

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Originally Posted by desiresjab

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

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

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Originally Posted by desiresjab

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.

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

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Originally Posted by desiresjab

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.

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

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Originally Posted by desiresjab

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.

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Is there an upper bound on this discrepancy?

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Originally Posted by desiresjab

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

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

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Originally Posted by YesNo

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.

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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.

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

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.

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.

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.

Maybe you need to study one of the other QR proofs to help stimulate new ideas?

The two sets of quadratic residues for p and q are: {1 4 2} {1 4 9 5 3}.

The first set actually represents the height of the rectangle, though p in the final result is represented by the lower half of WAXY, and q the upper half, and q represents the width.

(1, 1) is the first pair, and we will take those to be coordinates. That one is easy to assign a point. (1, 4) is next, and we take those to be coodinates, too, and so on and so forth.

* * * *
(1, 1) (1, 4) (1, 9) (1, 5) (1, 3)

(4, 1) (4, 4) (4, 9) (4, 5) (4, 3)
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(2, 1) (2, 4) (2, 9) (2, 5) (2, 3)

Only those pairs with an asterik have unique coordinates in WAXY, the other pairs simply reduce to duplicates. See a pattern? Remember, the first coordinate is reduced (mod 3), and the second is reduced (mod 5). And do not forget!! our first coordinates above are vertical coordinates, and the second coordinates are the horizontal. That is the opposite of what we are all used to from algebra where the horizontal x-coordinate is always the first element in the ordered pair and the vertical y-coordinate is the second element.

If I reversed the order of the digits in the ordered pairs, we still get eight ordered pairs with asteriks, as long as I reverse the moduli too.

Were eight asteriks a coincidence above, or will there always be exactly the same number of pairs with natural coordinates as there are lattice points in one of the triangles of WAXY?

Only more grinding will know.

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Originally Posted by YesNo

Maybe you need to study one of the other QR proofs to help stimulate new ideas?

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This is a new idea, lad. It takes quite a lot of labor to investigate one idea that is already deep in, as you can see from my last post, which dealt with only one pair of primes, 7 and 11. I am so scattered around I cannot do it all in one day. But an amateur is having fun.

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Originally Posted by desiresjab

Were eight asteriks a coincidence above, or will there always be exactly the same number of pairs with natural coordinates as there are lattice points in one of the triangles of WAXY?

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I like how you paired the residues. I hadn't thought of them in that way before.

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Originally Posted by YesNo

I like how you paired the residues. I hadn't thought of them in that way before.

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I hadn't either. Last night I did more numerical experiments. Those pairs that work directly as coordinates within WAXY without being reduced, and those that do not, always seem to partition the two halves indentically as the diagonal does, but I can no longer conjecture the larger half will always be the one expressible as coordinates. For 5 and 11, WAY and YAX contained 6 and 4 lattice points respectively, but only four expressions for lattice points out of the ten total. There goes that one. The expression went with neither the larger half nor the one on top. At least we know the one on top will always have as many or more points than the bottom, since it wil always get the point (1, 1). Just the lean of the diagonal because of our convention of always making the short side the vertical dimension gets that done. Because we can remember a few conventions and have wiki-pejia access with Eisenstein's rectangle, we can communicate more easily. Tug lines are good in deep water.

More might come of investigating residue pairs further. It could also be a deadend that dispalys a lot of not unlikely connections without helping to get deeper into the process. Both are true of any investigation, however, and since I do not see any other method that might possibly lead ahead right now, I will roll it around for a while, as usual, without doing any more work.

If I can figure out what the number of natural expressions represents, that might be used as another means of going deeper into the machine.

I suppose the thing to look at now is the distribution of the "natural points" within WAXY, besides (1, 1), which is always predictable, and see if they tell me anything new or are simple reflections of ideas I already know.

One thing jumps out--the pairs that make it will be 1-heavy, with both coordinates small. The 1 column and the 1 row should be more full than the others.

The residue pairs seem like an interesting approach. I don't know enough about the topic to know if anyone else has looked into this. It may lead to some other unexpected results.

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Originally Posted by YesNo

The residue pairs seem like an interesting approach. I don't know enough about the topic to know if anyone else has looked into this. It may lead to some other unexpected results.

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There are many experts right now who could tell us exactly where pairing residues leads. I never even pretend I might out think them or dream up an approach no one else has tried. The smartest people in math are simply too smart. Questions we have to dig in for some numerical on, they solve accurately in their visions. It is probable that if this leads anywhere we will end up in the territory of other maths we know nothing about, speaking in our usual language.

All the "unnatural pairs" pairs we generate with our multiplication are in reality duplicates of the natural pairs. I feel strongly that the natural pairs have some other quality not possessed by the pairs that have no representation at all. I could almost put it into words right now, but they would be only words with no mathematical help yet. It is a matter of plugging both sets in somewhere and seeing the difference in their behavior.

Now wouldn't it be a shade of wonderful if it was something as simple as they are divided according to those which are residues two ways and those which are quadratic residues only one way? Now I just have to decide how I am going to decode it. But how does that work if the division is something like 6 and 4, as it is with primes 5 and 11? I am learning not to box myself in with conjectures.

For now we can feel confident assuming we know the pairs will always partition out the same, through the coordinates or through the diagonal division, but are unable to yet say why or how. Euler might be giggling at us right now.

* * * * *

People might wonder how we can spend so much time on quadratic reciprocity without being even dumber than we admit to, when college juniors and seniors take classes in number theory where it is presented and pass their tests. It was just one more class in a long string of difficult classes for college students majoring in math or science.

Do they really understand it? They pass the tests, I believe we would pass them, too. Furthermore, we would amaze many, probably including the professor, with how thoroughly we understood many aspects of the theory.

I took calculus and differential equations, and I did better than merely pass those classes. What you learn in such classes is how to operate the formulae and which ones to use when. Normal college classes are not about in depth looks at particular problems, but learning each new language and how to operate in it. None of them go away with a greater understanding than we have right now.

As a graduate student in math they might confront this problem full on in a class, once they have had both group theory and abstract algebra. Those proofs are from the eagle's perspective, many levels above our ground level and subterranean approaches. Their view tells them something like the crankshaft will not turn, at any rate, if the carbeurator is disconnected. It does not take them down in the engine where the numbers are. At best, they learn how certain classes or fields behave, which is where we may end up yet.

The 38 lectures I watched on abstract algebra did not prove QR. A few times it seemed like they were getting close to the same ideas, though. They did prove quite a few other propositions, however, including Fermat's little theorem. The thing that knocks you off your feet is how brief the proofs are. A few sweeps of the chalk and they are done. That is how high they are soaring.

* * * *
(1, 1) (1, 4) (1, 9) (1, 5) (1, 3)

(4, 1) (4, 4) (4, 9) (4, 5) (4, 3)
* * * *
(2, 1) (2, 4) (2, 9) (2, 5) (2, 3)


Here are the quadratic pairs for 7 and 11. On this word processor, the asteriks may not line up where I want them. They didn't. The point is, if you take the moduli back to three and five, those fifteen points reduce back to only eight points. Only eight points are represented above. There are more lattice points in WAXY, that have no representation at all. Let us track down the other seven.

(1, 2) (2, 2) (3, 3) (3, 2) (4, 2) (3, 1) (3, 4)

the point (4, 2) is not actually in WAXY. It is too large. Reduced all the way it is merely the point (1, 2)

Wait again. (3, 5) is a point reperesented nowhere. (4, 2) can be reduced by the smaller moduli, (3, 5), cannot, except to (0, 0), which mamma don't allow none of around here.

(3, 5) must replace (4, 2) in the list of pairs above. Here it is:

(1, 2) (2, 2) (3, 3) (3, 2) (3, 5) (3, 1) (3, 4)

We have their names, now what can we investigate or interpret?

What can they not do that the others can?

* * * *
(1, 1) (1, 4) (1, 9) (1, 5) (1, 3)

(4, 1) (4, 4) (4, 9) (4, 5) (4, 3)
* * * *
(2, 1) (2, 4) (2, 9) (2, 5) (2, 3)

Only eight distinct points are represented above, if you use the reduced moduli. All are valid, they just happen to lie outside of WAXY on the coodinate system, though of course they are within the larger ABCD, to keep things in perspective. Magically, they all reduce back to natural pairs already listed within the matrix. There are more lattice points in WAXY. Let us track down the other seven.

(1, 2) (2, 2) (3, 3) (3, 2) (3, 5) (3, 1) (3, 4)

Here are the two residue sets again. {1 4 2} {1 4 9 5 3}.

We intuitively understand why the point (1, 1) is represented. Can we understand why a point like (1, 2) above is not? We want to understand it in a better way than just that the two sets when combined in the order shown, cannot produce that pair. What do we need to try?

For (1, 2)
1 is a residue of 7, but 2 is not a residue of 11. That is a start. Keep going.
For (2, 2)
2 is a residue of 7, but 2 is not a residue of 11.
For (3, 3)
3 is not a residue of 7, but 3 is a residue of 11
For (3, 2)
3 is not a residue of 7, and 2 is not a residue of 11
For (3, 5)
3 is not a residue of 7, but 5 is a residue of 11
For (3, 1)
3 is not a residue of 7, but 1 is a residue of 11
For (3, 4)
3 is not a residue of 7, but 4 is a residue of 11

Remember, these are pairs which are coordinates, but were not generated in our combinatorial multiplication. There is at least one-way rejection, for the entire list. But for the matrix of pairs above with fifteen pairs, both never reject, all the time. Even the pairs that lie outside of WAXY are valid, they are just not inside WAXY, which is true for some of the quadratic residue pairs generated in the multiplication. Since some of the valid pairs generated in the combinatorial multiplication lie outside of WAXY, we can only expect some of the points inside WAXY to be valid residue pairs. In fact, eight of them are, and seven of them are not. This accords exactly with the split of the diagonal, and the split of the natural expressions.

Those lattice points within WAXY that aren't two-way accepting, fill the second column and fifth rows, exclusively, forming a right-heavy bar on the capital T of its shape. This puts 5 of the 7 points in WAY, but 2 of them in YAX. Those seven are the same ones we said earlier had no “Natural expression.” Natural expressions accompany pairs of two-way acceptance only. We wonder on the side if only one is usual.

For primes 7 and 11 and the rectangle WAXY, eight of the lattice point coordinates (the same ones with natural expressions) do indeed have two-way acceptance pairs as coordinates, and seven have at least one rejection. A perhaps complexifying aspect here is that one of those seven pairs (3, 2) is two-way rejecting.

When you represent the pairs as coordinates when Eisenstein's rectangle ABCD is on coordinates, then eight of the fifteen pairs of coordinates within WAXY are coordinates where both elements are residues of their respective prime in the ordered pair. Those are exactly the same pairs that have natural expressions that were generated in the combinatorial multiplication. Six pairs of coordinates have only one quadratic residue, and one pair (3, 2), has none, i.e. 3 is not a residue of 7 and 2 is not a residue of 11. Graph-wise, (3, 2) is where the leg of the T and its bar intersect and occupy the same lattice point.

Somehow, the diagonal and the value of the number of natural expressions, manage to cut WAXY correctly in terms of two numbers to be used as exponents, innocent lattice points, and actual coordinates. The diagonal does not cut all the naturals to one side, but gets the numerical partition correct. The T marks the exact positions. That coordinates even apply is great!

7 and 11, being a pair of 4n+3 primes, are never mutal hosts to one another, but some of the pairs generated are, in a sense. We generated the pairs the old fashioned way—through a multiplication process more basic than the one taught in grade school, and found out that 15 is the number of distinct pairs generated from two sets of 3 and 5 members, respectively. Eight of these pairs are mutally accepting pairs, and seven of them are not.

Now it's back to the think tank.

As you mentioned math classes are more to show students how to use the language and solve some problems. Generalizing does take one away from the details.

I was looking at the Sierpinski problem recently since that is the one that I have a computer working on for PrimeGrid. I am trying to see if I can get Python and MySQL to help generate covering congruences for some numbers that are unknown whether they are Sierpinski numbers or not.

A Sierpinski number is a number of the form k 2ⁿ + 1 where k is odd. A Sierpinski number is a k such that for all n none of the numbers are prime.

Paul Erdos made some discoveries concerning covering numbers. The Sierpinski gasket is a fractal object.

5 and 17 are a strange pair. They are both 4n+1. Phi/4 is 16, so there are plenty of factors of two left. Yet when the diagonal divides them it partitions them to 9 and 7. Since they are mutually rejective with that many factors of two left, that is all the diagonal could do.

Sometimes the diagonal is forced to parttion an odd number into two odd ones, as in 7 and 11, when there are only two factors of two (the minimum), or break an even one into two odds of different value, such as for 5 and 11, where there are only three factors of 2. But this is the first time I have seen a highly even number partitioned unevenly. The diagonal always gets it right. I have not had time to check the "natural" pairs for for 5 and 17 yet to see what they say. There are only 16 of them, so it will not be difficult.

Quote:

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Originally Posted by desiresjab

5 and 17 are a strange pair. They are both 4n+1. Phi/4 is 16, so there are plenty of factors of two left. Yet when the diagonal divides them it partitions them to 9 and 7. Since they are mutually rejective with that many factors of two left, that is all the diagonal could do.

Sometimes the diagonal is forced to parttion an odd number into two odd ones, as in 7 and 11, when there are only two factors of two (the minimum), or break an even one into two odds of different value, such as for 5 and 11, where there are only three factors of 2. But this is the first time I have seen a highly even number partitioned unevenly. The diagonal always gets it right. I have not had time to check the "natural" pairs for for 5 and 17 yet to see what they say. There are only 16 of them, so it will not be difficult.

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I did check those sixteen pairs of coordinates, and any conjecture crashes between the number of naturally expressible coordinates and the partitioning of lattice points. Those coordinates are merely the ones found strictly within WAXY. Only four coordinates out of sixteen are of this type for the two primes 5 and 17, and we know that number does not correspond to either value of the partition.

With certainty, we know that the diagonal for two 4n+3 primes will "thread," through an odd number of lattice points to partition them into one or other of WAY or YAX in unequal odd numbers.

With certainty, we know that a 4n+3 prime and a 4n+1 prime of lowest evenness for its kind (only divisible by 2 twice), will "thread" through an even number of points, partitioning them into odd halves which may be either equal or unequal, as far as we know.

With certainty, we know the two situations above were forced by a limited number of factors of 2.

In the Eisenstein rectangle lattice point graph for the two primes 5 and 17, the "lean" of the diagonal steals away three lattice points on the bottom row, but two of them get made up somewhere.

Whether or not the values of WAY and YAX can ever differ by more than two, becomes an interesting question in itself.

You might try constructing proofs of the certain items if for no other reason than to get a foundation for future results. There is a theory of lattice points that I am unfamiliar with that might be a place to start.

I thought you found examples where the WAY and YAX differed by more than two lattice points. Perhaps not.

I'm working on a problem at the moment and trying to get Python to generate some examples or a solution. The claim is that the sum over n starting with 1 of (n-1)ⁿ is never prime. So the goal is to find a prime in that sequence of integers or find a covering congruence to show that the sequence has no primes.

I'll be happy if I can get a script to generate composites of this form up to n = 1000. I think I have a workable algorithm for that, but I don't even have a way to show that a covering set of primes actually covers all of the numbers in the sequence.

Quote:

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Originally Posted by YesNo

You might try constructing proofs of the certain items if for no other reason than to get a foundation for future results. There is a theory of lattice points that I am unfamiliar with that might be a place to start.

I thought you found examples where the WAY and YAX differed by more than two lattice points. Perhaps not.

I'm working on a problem at the moment and trying to get Python to generate some examples or a solution. The claim is that the sum over n starting with 1 of (n-1)ⁿ is never prime. So the goal is to find a prime in that sequence of integers or find a covering congruence to show that the sequence has no primes.

I'll be happy if I can get a script to generate composites of this form up to n = 1000. I think I have a workable algorithm for that, but I don't even have a way to show that a covering set of primes actually covers all of the numbers in the sequence.

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I may have stated incorrectly once I had an example. I believe I have no examples of WAY and YAX with a difference of more than two lattice points. These problems get huge to generate by hand with relatively small primes. I have no mathematical software to assist.

Your problem sounds interesting, and either has an echo or a false echo of Fermat.

It is a snap to show that (p-1)(q-1)/4 is equivalent to Euler's totient function. Pretend that these two primes are really, really, really huge. We can always tell their types, but ascertaining whether one is a quadratic residue of the other may be next to impossible by hand. What can we do?

We also pretend we have a computer capable of multiplying (p-1)(q-1), in fact we will need one. Trusty division by four is our next step. Now we have something to look at. We can tell the species of this quotient, too.

If the quotient is already an odd number, we know the diagonal will produce an even and an odd number. We must have been dealing with two 4n+3 primes.

If the quotient is a 4n+3 number, at least the possibility if not the certainty of (1)(1) is preserved, though I see no way yet to determine if it is that or (-1)(-1) for very huge numbers.