Cosmology

[desiresjab]

Eisenstein has done something extraordinary. His proof actually has nothing to do with QR, other than a method for building the right exponent to go on -1, though it does involve the primes p and q. It looks at the ratios of p and q when the prime in the numerator is multiplied by successive even numbers under the modulus of the other, with a chop function appended. In other words a function that always rounds down instead of moving to the nearest value. It would have been sufficient to find the correct parity under any circumstance, but Eisenstein is more exact than that, producing the exact exponent on -1 as the number of lattice points in the prescibed regions of his p by q rectangle. Only the (p-1) by (q-1) part interests him, containing the interior lattice points of the larger rectangle, and then only those with even coordinates.

This proof is wonderfully clever. The downside is that it will not reveal any deeper properties of numbers that help elucidate why QR works. Deeper investigations might uncover why it works. It proves what it proves--that Eisenstein's method will always find the right exponent for -1 in the Legendre symbol.

[YesNo]

Is the computation done in polynomial time in terms of the number of digits of p and q or something slower?

[desiresjab]

Is the computation done in polynomial time in terms of the number of digits of p and q or something slower?

Uhhhh....I am not sure I understand the question.

Finding a quadratic residue of a given number no matter how large, should be a P time exercise. You may be referring to something like RSA encryption or Diffe-Hellman key exchange. Various encryption systems are based on particular aspects of number theory laws. It can be quadratic residues, primitive roots or mod inverses they use to "conceal" the message.

In reverse the problem gets nasty real fast, as in NP nasty. It is easy to give some quadratic residues of a number q. But given a quadratic residue, it is impossible to find q in ploynomial time, if the numbers involved are long enough.

Currently, it takes numbers of about four thousand digits length to get your bank account information encrypted securely. If some country or individual had the power to quantum compute, it has already been mathematically proven they could break our present codes in seconds.

[YesNo]

That answered the question. I was wondering if there were a computation problem still unsolved. I suppose a quantum computer runs faster because of a potential parallel processing involved. Wouldn't a network of computers working in parallel be able to simulate such a computer?

[desiresjab]

That answered the question. I was wondering if there were a computation problem still unsolved. I suppose a quantum computer runs faster because of a potential parallel processing involved. Wouldn't a network of computers working in parallel be able to simulate such a computer?

That is the same question I have had. I have a hunch the answer would be "yes," provided that we link enough silicon computers together to fill the solar system, or some other great volume of space.

I went to town today and forgot graphing paper. If I graph out some p's and q's Eisenstein style, another key to QR may pop out visibly or algebraically.

I have some other studies I am stalling right now because I cannot let go of QR when I am so close. Like one of those movie bounty hunters who has pursued a particular fugitive for a long time, I cannot go for coffee now that the fugitive has been sighted. However, at my age I find I need two days rest after pursuing the most intense thinking for one day. Total forced mental focus was something I learned in math and then adapted for dummies from reading about Newton and Tesla. Now, in math I have to force it, because it hurts me to concentrate that hard on x, y and z in a foreign medium, whereas in a field like fiction writing I can stay immersed indefintely without forcing myself to, indeed without ever becoming conscious of the need to force myself to do anything.

The creative process is so much more enjoyable while it is happening, but when titantic struggles with x, y and z are over and have resulted in a surrender without terms, I find my picture of the universe is altered, and my picture of myself, as well. In this medium of math I am no natural, but I am a zealous convert.

[Dreamwoven]

I am lost in this discussion, never was any good with maths. This discussion of time-warps and the past and future of Space-Time may be of interest: http://www.space.com/31495-space-tim...ekly_2016-1-04

[desiresjab]

I finally reeled Eisenstein in all the way. Of course that does not mean I understand Eisenstein as well as Eisenstein did. I feel like champagne anyway.

[YesNo]

Interesting survey article on spacetime, Dreamwoven. In particular this quote:

"It might be that space-time at very short distances takes yet another form and perhaps is not continuous," Amendola said.

Congratulations, desiresjab! I was reading this about quadratic reciprocity. It is a very elementary summary of it: http://sites.millersville.edu/bikena...-residues.html

The part I was interested in was if a prime is a primitive root for another prime it would have to be a quadratic nonresidue. I haven't tried to understand Eisenstein's proof.

[desiresjab]

Interesting survey article on spacetime, Dreamwoven. In particular this quote:

"It might be that space-time at very short distances takes yet another form and perhaps is not continuous," Amendola said.

Congratulations, desiresjab! I was reading this about quadratic reciprocity. It is a very elementary summary of it: http://sites.millersville.edu/bikena...-residues.html

The part I was interested in was if a prime is a primitive root for another prime it would have to be a quadratic nonresidue. I haven't tried to understand Eisenstein's proof.

You are welcome. As usual, one wonders why he didn't see it sooner. I guess it is the leap from the quadratic to the linear. -1 is a dummy base, only good for determining if the exponentiated value is positive or negative. His whole lattice graph is linear, yet it explains a quadratic law. People like Eisenstein are not really human, they just share some genes with the rest of us.

I am going to toy around with making a new encryption system. If nothing else, I will learn the present systems better. I have to come up with an appropriate function.

* * * * *

There is no guarantee that quantum theory and Einsteinian physics are uniteable. Parts of Einstein are already dated, Ed Mitchell has said. What if never the twain shall meet?

[YesNo]

If those quantum computers ever happen beyond a few qubits we may need a new encryption system.

[desiresjab]

If those quantum computers ever happen beyond a few qubits we may need a new encryption system.

Yes, and it is hard to even imagine what it might be. I have been kicking around some ideas for a system that would be easily patentable. The normal trick is to multiply two huge primes p and q together to produce n, which will be part of the modulus. Pick a number e relatively prime to n as an exponent to encrypt the message, as in M^e. Then you find the inverse of e (mod φ(n)). It is this e^-1 which will be used to untangle the message on the other end. You cannot find φ(n) without knowing the factors of n. That is the immense diffculty they impose on hackers--they have to find φ(n) to get anywhere.

I am looking for a system that does not rely on factoring. Encryption is one area of math with a big, big future. I have a few wild ideas.

[YesNo]

Maybe using larger primes will keep the current methods going until something better comes along.

I'm still putting the pieces together on the Artin conjecture. A simpler question would be "Given a number m, are there infinitely many primes p for which m is a quadratic nonresidue?" This would be a larger set since a quadratic nonresidue does not have to be a primitive root such as 8 or 12 mod 19 unless I calculated it wrong.

[desiresjab]

Maybe using larger primes will keep the current methods going until something better comes along.

I'm still putting the pieces together on the Artin conjecture. A simpler question would be "Given a number m, are there infinitely many primes p for which m is a quadratic nonresidue?" This would be a larger set since a quadratic nonresidue does not have to be a primitive root such as 8 or 12 mod 19 unless I calculated it wrong.

One aleph is as big as the next aleph.

[YesNo]

I thought Aleph one was strictly bigger than Aleph null.

[desiresjab]

I thought Aleph one was strictly bigger than Aleph null.

Otherwise known as Aleph nought, as well, the "smallest" infinity. They are the same entity, all of them, as far as I know.

To make all possible sets from a set of size n, take 2ⁿ. The next set after aleph, is 2 raised to the aleph power. You can repeat the process over and over getting bigger sets. No one knows which of these powers has the power of the continuum.

There is an infinite heirarchy of infinities, each theoretically greater than the next, but we only have examples of two kinds. One can be thought of as the counting numbers, and the other is the uncountable points on a line, i.e. the irrataional numbers, more specifically the transcendental numbers.

The former do indeed have the cardinality of aleph, and the transcendental numbers may have the cardinality of the continuum. I believe the latter is not known for sure. Kronecker would certainly disapprove.