Cosmology

Let me know if this is the wrong place...

"The superfluid Universe":

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We are used to thinking that quantum physics dominates only the microscopic realm. But the more physicists have learned about quantum theory, the more it has become clear that this isn’t so. Bose-Einstein condensates are one of the best-studied substances that allow quantum effects to spread widely through a medium. In theory, quantum behaviour can span arbitrarily large distances, provided it isn’t disturbed too much.

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https://aeon.co/essays/is-dark-matte...tm_source=digg

... and a toon: http://www.gocomics.com/bloom-county

Ta ! (short for tarradiddle),
tailor STATELY

I'm on this list and though I don't understand all the mathematical stuff the main players here write, I enjoy the discussions between desiresjab and YesNo. Read the thread and you will see what I mean.

I was offline for half a week with computer issues. The only thing to do was to go to pen and paper.

What do we know about squares in general? What is one fact of odd squares?

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Originally Posted by desiresjab

What do we know about squares in general? What is one fact of odd squares?

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I don't know except the obvious in the integers, Z, that one gets an odd number when an odd number is squared and and even number when an even number is squared.

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Originally Posted by tailor STATELY

Let me know if this is the wrong place...

"The superfluid Universe": https://aeon.co/essays/is-dark-matte...tm_source=digg

... and a toon: http://www.gocomics.com/bloom-county

Ta ! (short for tarradiddle),
tailor STATELY

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I liked the cartoon.

I didn't know that the theories around dark matter had two major variations, those promoting modified gravity and those promoting a cold particle. The superfluid idea is also new to me. Maybe it can bridge the ideas. The modified gravity reminds me of a talk by Rupert Sheldrake where he questioned whether physical constants, in particular G, were actually constant, but changed. If G changed that would be one way to get modified gravity.

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Originally Posted by YesNo

I don't know except the obvious in the integers, Z, that one gets an odd number when an odd number is squared and and even number when an even number is squared.

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I thought a very simple fact might be surprising. All odd squares are 4n+1 numbers. There are no 4n+3 squares; such an animal cannot exist.

An important fact for anyone trying to learn modular arithmetic has to do with symmetry. In normal arithmetic the negative and positive integers have symmetry across the point zero on the number line. That is, the absolute value of -5, for instance, is equal to 5. In modular arithmetic with primes, this is no longer true--

-5≡6 Mod 11.

Perfect multiples of the modulus have familiar symmetry across zero, but no other residue class does.

Under any prime modulus p, start squaring the positive integers n in succession. The series will always begin with the standard squares you are familiar with...1,4,9,16..., until n becomes greater than p^1/2 (the square root of p), at which point n² will wrap around the modulus to some value.

Quadratic reciprocity is about how two moduli wrap around each other under quadratic pressure.

We may further notice that when both p and q are primes of the species 4n+3, their squares always wrap around the other modulus so that p²≡2 (mod q) and q²≡1 (mod p), or vice versa. They will always express this relationship when squared within the modulus of the other. This is a fact of the universe, as inviolate as 2 is the successor of 1.

One always checks first to see if the larger of p and q simply reduces to a familiar square under the other as modulus. If so, the work is done. Otherwise one starts squaring n's to see if any wraps around to the value of p, under q as modulus, or vice versa.

Not surprisingly, then, two 4n+3 primes p and q wrap around each other just as their squares do. Either 4p+3≡1 (mod 4q+3) and 4q+3≡2 (mod 4p+3), or vice versa.

We must remember that if a≡b (mod m), then a^p≡b^p (mod m).

It appears my recent posts on 4n+3 only apply if one of the 4n+3 primes is 3 itself. More later. Three is not typical. Or is it?

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Originally Posted by desiresjab

I thought a very simple fact might be surprising. All odd squares are 4n+1 numbers. There are no 4n+3 squares; such an animal cannot exist.

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That is a more interesting fact than the one I presented. Here's my proof of it since it is not immediately obvious:

Assume there exists an odd square m² congruent to 3 mod 4 to get a contradiction. There are two cases to consider: m is either congruent to 1 mod 4 or m is congruent to 3 mod 4.

Consider the first case, m ≡ 1 (mod 4). Then there exists r such that m = 4r + 1 and m² = (4r + 1)(4r + 1) = 16r² + 8r + 1 which is congruent to 1. So m is not congruent to 1.

Consider the second case m ≡ 3 (mod 4). Then there exists s such that m = 4s + 3 and m² = (4s + 3)(4s + 3) = 16s² + 24s + 9. Since 9 is congruent to 1 mod 4, m is not congruent to 3.

In all cases m is not congruent to 3 mod 4 and since this contradicts the assumption, the assumption is false.

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Originally Posted by desiresjab

We may further notice that when both p and q are primes of the species 4n+3, their squares always wrap around the other modulus so that p²≡2 (mod q) and q²≡1 (mod p), or vice versa. They will always express this relationship when squared within the modulus of the other. This is a fact of the universe, as inviolate as 2 is the successor of 1.

One always checks first to see if the larger of p and q simply reduces to a familiar square under the other as modulus. If so, the work is done. Otherwise one starts squaring n's to see if any wraps around to the value of p, under q as modulus, or vice versa.

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I would say it is a fact of the axiom system and the set of elements one is using rather than a fact of the universe. One could change the axiom system or the set of elements and get something different. For example, Euclidean geometry need not have much to do with space in the universe around us, but the results would be inviolate facts within the axioms of Euclidean geometry. Only if one can't consistently change the axioms would it be possible to look at the results as relevant to the universe.

I agree that the computationally hard part comes from the wrapping process.

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Originally Posted by YesNo

I would say it is a fact of the axiom system and the set of elements one is using rather than a fact of the universe. One could change the axiom system or the set of elements and get something different. For example, Euclidean geometry need not have much to do with space in the universe around us, but the results would be inviolate facts within the axioms of Euclidean geometry. Only if one can't consistently change the axioms would it be possible to look at the results as relevant to the universe.

I agree that the computationally hard part comes from the wrapping process.

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You can do these things anywhere. Euclidian geometry would be an outside geometry in some universes. Its laws would remain true, just as the laws of non-Euclidian geometries are true for us.

The wrapping process of moduli can turn a 4n+1 square into a 4n+3 number since, for instance three is a square under some moduli.

I only need to pinpoint the mechanics that force (4n+3)² to perform its consistent behavior under prime moduli, for the whole thing to shake out. It is a matter of mechanics. A mechanical detail is eluding me so far. That detail will clear up every question. I not only sense this is true, I know damned well it is. There is no doubt, either, that that detail is clearly available in group theory, which is why so many proofs rely on it.

I still believe it is something I can get from peering at the numbers. My investigations are going deeper underground where I need paper and pencil.

Remember the Martinson list for the sums of squares? He mentioned that any prime number generated by a sum of two squares was a 4n+1 number. He mentioned that the table would generate every prime of 4n+1 makeup. He did not mention that all odd numbers in the table were also 4n+1 numbers. Close inspection reveals that sums of two squares can only be 4n, 4n+2, or 4n+1 numbers.

On another note of interest. Breaking a large 4n+1 prime into its unique sum of two squares, is every bit as difficult as factoring. I have not delved deeply enough, but I wonder if any of the present encryption systems are taking advantage of this. A new function as the basis means no patent battles.

Anyway, I feel I am very close to the final solution with QR. I know where to dig and I think I know how to do it.

Here is a curious fact about 4n+3 primes. Look at seven and its squares with regard to other 4n+3 primes.

Because of what we already know, we can state unequivocally that no 4n+3 prime greater than 49 can ever wrap back to be a square (mod 7). Why? Because 7² is a normally occuring number (mod 59) and it will be the square between the two, since there can always and only be one square between 4n+3 primes.

7² is the naturally occuring quadtratic residue of every 4n+3 prime larger than 47, not the other way around, ever.

This idea has its way of working with 4n+1 primes and mixed couples, too. If the larger prime does not reduce back to a sqaure under the smaller prime, then the smaller one will not stretch to a sqaure either, by the rules.

Therefore, under any moduli if p²<q, p² is always a natural residue, meaning there is no wrap around by the sqaure.

Among 4n+1 primes and mixed couples, this fact forces them to both be squares, and to never not be mutual squares, since they must act the same way as each other. There is only a question whenever q is < p², or vice versa. Otherwise, the results are automatic. Of course, we still must explain why they behave the way they do when p²<q. It is always good to see the task more clearly.