My posts are disappearing.
My posts are disappearing.
My own unsloved problem is Brocard's problem. In mod arithmetic it might be stated thus:
q2≡1 (mod (p!)). Things that look as if they should be simple, turn out to be near impossible.
That's true. The exponent (p-1)/2*(q-1)/2 are the number of lattice points within the triangles AYX and WAY. However, the Gauss Lemma comes up with a smaller exponent. My difficulty is how to show that the smaller exponent can be replaced by the larger one so that the calculation depends only on p and q.Originally Posted by desiresjab
For example, let p = 11 and q = 13. Then (p-1)/2 = 5 and (q-1)/2 = 6. To see what the Gauss Lemma provides consider the numbers from 1 to (13-1)/2 = 6 or {1,2,3,4,5,6} and multiply them by p = 11 mod 13. This gives {11,9,7,5,3,2}. Values over 6 could be viewed as negative if we used the residue set between -6 and 6 mod 13 rather than the one between 0 and 12 mod 13. There are 3 values larger than 6, namely, {11,9,7}, and -13 = -1 = 11(13-1)/2 mod 13 = 116 mod 13. That would be (p|q) = (11|13) = -1. The Gauss Lemma states that is another way to calculate (p|q) rather than to raise p to the (q-1)/2 power mod q.
Considering (13|11), we would look at this set of residues mod 11 {1,2,3,4,5}. By the Gauss Lemma we multiply each of them by 13 and get the following {2,4,6,8,10}. Now we count those in the set greater than 5 = (11-1)/2 and find there are 3 of them. So -13 = -1 = 13(11-1)/2 mod 11 = 135 mod 11. That would be (q|p) = (13|11) = -1.
Since p = 11 = 3 mod 4 and q = 13 = 1 mod 4, quadratic reciprocity says that (p|q) = (q|p) which turns out to be the case since they both equal -1.
My problem is, I understand the proof of Gauss Lemma which gives exponents of 3 for both p and q. But I don't see how either Gauss or Eisenstein raises that exponent to 5 and 6. I can see why they would want such exponents. It would would be easier to calculate.
My blog: https://frankhubeny.blog/
5 and 6 are the effective dimensions of the smaller rectangle comprised of the triangles AYX and WAY. These match up with Euler's criterion. These are the ones you want, I believe. Adding 1/2 to their dimensions allows lattices points on the perimeter of the smaller rectangle with dimension 5 and 6. Therefore they are all in the interior of the rectangle augmented by 1/2 in its dimensions.
5 and 6 get the right product, but I do not see anything in the triangles denoting a significance of 5 and 6, other than their product and their opposite parity. I also see nothing which tells me about -1, other than a fact we already know--that -1 is a residue of all 4n+1 primes. Since we know that already about primes, it does not seem important to me that the diagram does not speak to that aspect.
It looks like the exponent of -1 in the following
(p|q) = (-1)(p-1)(q-1)/4 (q|p)
is just an algebraic way to write the English phrase
(p|q) = (q|p) unless p and q are both congruent to 3 mod 4 in which case (p|q) = -(q|p).
It also occurred to me the main reason for quadratic reciprocity is for calculation purposes. This theorem allows us to flip the p and q and then reduce the larger one.
My blog: https://frankhubeny.blog/
I'm not too sharp right now. I have come down with the flu or something. Reciprocity is easy to state in English. The first step to understanding it is to learn enough math (mostly modular) to understand the English.
Now of course it has a much bigger reason--the razzle dazzle of ciphers.
I'll be back after I throw up and sleep.
Get well!
The more I look at this the more I feel like I am still in the shallow end of the pool.
Edit: After reading a proof, different from Eisenstein's, I began to see how the primes p and q are connected: They are each related to their product, pq.
Also the proof depended upon something nearly obvious: if three integers are added together and their sum is an even number then either all three of the integers are even or only one of them is.
http://www.lehigh.edu/~shw2/q-recip/gauss5.pdf
Last edited by YesNo; 01-28-2016 at 01:05 PM.
My blog: https://frankhubeny.blog/
Back with better health.
How Gauss determines that what me might call the overflow values are enough to determine the quadratic relationship is algebraic magic, of course, but not necessarily transparent. It gives another fact, but the torch in the fact is hard to light. There is extreme familiarity with modular operations and the Chinese remainder theorem and how they all apply. Seeing how every rule you need applies at the right time and place is always going to be the case. In one's personal investigations, if one misses one of these, a great deal of time can be lost but not necessarily wasted in chasing down proof along the way of something that boils down after all to a basic law of modular arithemtic the explorer has not yet assimilated into his mathematical vocabulary fully enough so that its applications and assumptions come naturally as they do in normal algebra.
No matter how simple they continue to try to make QR, it always turns out pretty complex, except in terms of the laws themselves, which are clear and easy to apply. In asking why the two species of primes themselves behave the way they do with themselves and with the other species, it is profitable to remember that they only do so in modular arithmetic, where QR is a theorem. The comparison to normal arithmetic in the Martinson link I gave earlier is still the most illuminative and suggestive article I have seen yet. For that reason, I believe a good review of Fermat's and LaGrange's sums of squares is in order. I did this in cursory fashion a few months back, without settling in for the full ride with different hosts.
Another nagging propositon I looked at only a few weeks ago is Bertrand's paradox. It deals with geometry and the power of the continuum infinity of points on the surface of a sphere. Someone gives a good explanation on YouTube of how it is mathematically possible to dissect a sphere of diamter X and reassemble the parts into two full and complete spheres of diameter X without adding any new material. No man or machine could actually make these slices and chops, but in theory it is feasible. Or is it an unresolved paradox of infinite set theory? It deserves a second dip. So many things to chase down.
Last edited by desiresjab; 01-29-2016 at 01:55 PM.
To continue in the same vein, the quote But this is just the set of integers, lifted directly from the end of the Gauss proof is, in English, what always happens at the conclusion of proofs in this mode of math, now isn't it? The results of operations on a residue system are shown to be equivalent to another set of integers previously defined. When all you need is parity to prove your point the sets do not even need the same cardinality to yield their information. This is heading toward set theory and group theory. I am still stuck on the idea of an easier vantage to peer at the heart of the law and see naked numbers bathing.
Deciding how one can show some particular set equals another set or subset must be the normal way to proceed then, what you strive to frame your question in terms of in modular forests. That is my nutshell observation.
I looked around on YouTube and found this description of Bertrand's Paradox: https://www.youtube.com/watch?v=uI2FnUmBeeo
It seems that the paradox is resolved once one defines what it means to "choose a chord at random". One of the choices started with a fixed point, another with a fixed diameter the chords had to cross and the third asked whether the midpoint of the chord was inside or outside an interior circle. Not all of the possible chords were permitted by the selection constraints in the first two examples. I suspect the third example did include all possible chords.
Last edited by YesNo; 01-29-2016 at 07:03 PM.
My blog: https://frankhubeny.blog/
Bertrand's problem looks harder than I realized: https://en.wikipedia.org/wiki/Bertra..._(probability)
I don't think the problem is resolved, as I claimed above, by saying there are more chords in one of the three examples. One can assume there is one chord in each. Then what is the probability that its side is longer than the side of an inscribed equilateral triangle. I think you are right in looking at this as a problem of picking a point from an infinite number of points.
Last edited by YesNo; 01-30-2016 at 10:12 AM.
My blog: https://frankhubeny.blog/
my recollection of time is that it began with the universe, (space-time). so the first thing we have to do is wrap our heads around the question: if space-time began with the universe, then how can we ask the question, "what was before?" before implies time, but it didn't exist.
A fascinating idea that i have wondered about is the Schrodinger's cat conundrum. if there has to be an observer before any event can take place, then there had to be an observer before the first two particles o matter inter-acted, did there not? This has led some to theorize that there was intelligence prior to matter, rather than the other way around.
The Schrodinger cat problem keeps bothering me as well. Basically, every time I think I understand what it is supposed to show, I doubt that I have it right. Although I haven't read much lately, I have Amit Goswami's "The Self-Aware Universe" on my desk. He promotes "idealist science" as opposed to "materialist science".
My current view is slightly different from "there was intelligence prior to matter". At the moment, I don't think unconscious matter exists. There is nothing but intelligence. What we see as matter is conscious at a lower level that appears unconscious at the macro level where we view it.
Of course, I might be completely wrong.
My blog: https://frankhubeny.blog/
Posting Permissions
Reply With Quote