Gaussian primes

Given 3 quadratics, is possible to solve cubic if the 3 are Gaussian integers?

If there is a map from 3 quadratics and the connected space exists, is there simple solution in terms of prime factors?
Further, is there a way to section the circle, with quadratics so that 8 cubics are as sections on the sphere?

This can be done with Kepler triangles when one of the circle's sections has one apex at (0,1), and two other equal-area sections with apex each at intersection of the half-circle, at y = -1, i.e. origin at (0,-1). Does this describe a hyperbolic manifold (does it represent occultation of, say, the lunar surface by the earth's?)

This is really astronomy, on paper.

 

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Try constructing triangle on the circle tangent to G, diameter GD.
Start with ABC is rt. triangle with AC = 2, AB = 1. Draw Kepler triangle on hypotenuse of ABC, project CB to D, connect C also to E, s.t. CED is also rt triangle and AC = EC = GC.

The two semicircles on origins C, and CD/2 meet at G.
Circle at O (radius r=1.25), on DE is "sphere" on Kepler surface, semicircle on CD/2 is geodesic, pi is 4th order irrational root of quartic.

Map ABC, CGD, CGE to Kepler triangle in unit square which is produced in 1/4 of circle as in first post (p1) above. Circle on O, and semicircle on O' at O - 1/2 in square reveal crescent or lenticular curve on Kepler surface, inside 2 (unit) squares (one above the other). Golden ratio, by Kepler, is at intersection of O' radius 'across' upper 1/2 of lower square, when rotated to y axis of first circle origin O.

If AB = 1, AC = 2, what are the sides of CGD in the construction?

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I spent 3 or 4 minutes trying to follow you, but can not. It would help is you made at least footnote to define or at least illustrate terms (like Kepler triangle, Gaussian primes, etc.) and removed your ambiguities. For example is your traingle ABC which is tangent to circle and contacting it at point G, entirely out side of the circle, or at least partially inside the circle? You give size of some items but not GD, (It could be 0.1 or 1000?) etc. You seem to know something that I could learn from so I am interested if you can learn to comunicate clearly.

 

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Ok, sorry yes ABC, CDE, GDC are all partly inside circle. The three right triangles are related by law of areas and I believe G is then a tangent bundle, on the lower part of the circle, which has radius >1.
ABC is not on tangent, but CG is tangent; DE bisects circle and DG is 'off-axis'.
.
The map S2 -> S3 is also constructible in unit circle/ square-in-quadrant and Kepler equal areas, and golden ratio so that pi is 4th order root. There is constructed diagram, courtesy of Greek mathematician, which I find here:
http://www.stefanides.gr/Html/piquad.htm

interesting because need only ruler and compass to see relation, diagram is like sphere rotating from 'H' in diagram, to 'J' on GD. 'HJ' is also geodesic on circle -> sphere.

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There is text below the drawing. It fills my screen and I do not know how to reduce it.

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You are still introducing terms many may not know with no definitions, not one yet. New ones include "law of areas" "tangent bundle" etc.
"Kepler's equal areas" does not refer to any triangle as it is the area contained by two lines and part of a curved orbit.

I know what the golden ratio is, an irrational number whose value is easy to derive with equation derived from the geometic interpretation. To illustrate what you are failing to do (to be clear) I will define it for others who do not.

If a rectangle has sides L and S (for long and short) and when one cuts out of it (at one end) a squared of side S an new rectangle with sides S & (L-S) but now S > (L-S) remains. If this new, smaller rectangle is the same shape as the original, i.e. such that L/S = S/ (L-S)* then it is a new smaller "golden rectangle" and L/S is the "golden ratio." From this new smaller rectangle another square can be cut, leaving still another golden rectangle, etc. forever making smaller squares and golden rectangles. The ratio, for which this is true, L/S defines the "golden ratio."

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*To find the numerical value of the golden ratio (about 1.4+ as I recall.) set S=1 and solve this equation for L.
Note If reader has never done this, find the ratio (wiki if too lazy to solve the equation) draw a large one accurately and make several smaller ones. Then put point in the center of the squares. These points all fall on a well know spiral, but I forget its name.

Now you please do the same for the terms you use (or give links that define them). IMHO, a main point of these forums is to educate. so make the effort, if you want to continued the discussion.

 

To post figure available in the internet:

Right click on it. Open "properties" then copy the address of the figure, which I'll now call XYZ.

Then to post figure, use: {img]XYZ[/img] but of course the leading { needs to be [

PS thanks for the Kepler triangle infromation. I had never heard of it and still do not know why Kepler made it but the wiki link you gave does tell how to make the golden retangle with it, using only the classic tools and I did not know to do that that either. (Why in last post I told reader to find the ratio and measure to draw one. I see or am reminded that (root 5)/2 is the golden ratio, (~1.11; My 1.4 from memory was wrong.)

 

Neat illustration in your post 8. It is a very early and limited example of a fractile in that as you step down and then scale back up to be able to see,
the pattern is unchanged.

Here is clear definition I found of Gaussian Primes:

“A Gaussian prime p is a Gaussian integer a+bi (where i is the imaginary unit and a and b are real integers) that is divisible only by the units 1, −1 , i and −i , itself, its associates and no others. For example, 3+20i is a Gaussian prime because there is no pair of Gaussian integers (besides the units and associates) that multiply to 3+20i . But 3+21i is not a Gaussian prime because 3(−i)(1+i)(1+2i)^2=3+21i . If a+bi is prime then so are a−bi , −a+bi and −a−bi , as well as the associates b+ai , b−ai , b−ai and −b−ai .

The real and the imaginary parts must be of different parity. For a real prime to be a Gaussian prime of the form p+0i , the real part has to be of the form p=4n−1 ; the same goes for the associates 0+pi . It follows from Fermat's theorem on sums of two squares that since real primes of the form p=4n+1 can be represented as x^2+y^2 , then in the complex plane they have the factorization (x+yi)(x−yi) . For example, 17=4^2+1^2 , so (4+i)(4−i)=17. “

From: http://planetmath.org/encyclopedia/GaussianPrime.html

BTW, there is a name for pairs of primes of the form p=4n−1 & p=4n+1 (and for n^2 + 1 & n^2 - 1) but I forget it too. (Thanks for naming the spiral for me.) I think it is known that an infinite number of them exist, but am not sure I am remembering this correctly. (Perhaps they are still trying to prove that?)

With Gaussian primes, we are getting a little too mathematical to hold my interest, but I am fascinated by these geometric facts. You know I bet that if the three pairs of opposite sides of an irregular hexagon, which is inscribed in a circle are extended to their three pair-wise intersection that these three intersection points always are co-linear (on one line.) - This is my favorite geometric fact. (I made and framed a nice illustration of it.) What is yours? I have spent a few hours trying to prove this true. I have also read that (Gauss, I think it was) proved it at age 12! Is that true and who it was?

I will probably drop out of this thread now, but there are some highly talented math guys here who can discuss whatever your point / question in the OP is.

Again thanks for educating me a little.

 

I'd be surprised if this isn't just another one of Reiku's sock puppets like his last one which just posted seemingly random combinations of buzzwords in the hopes of suckering in people who understand such things as 'star algebra' or the like into trying to explain them. Reiku seems to think the joke is on the people who know something if he can fool them into posting their knowledge in an attempt to correct his ignorance.

Just when you thought Reiku couldn't get any more pathetic.

 

In post 3, I noted that werzel has an inability to coherently express himself so I think you are correct. - There still seems to be no point to this thread, just cut and paste from searches.

I am very lazy, so seldom search, but Reiku likes to and often posts the results, even when his understanding is so low that he has posted mutually contradictory results for different sources without comment noting that.

None the less in this thread he found some things I did not know, such as how to construct the golden rectangle with only the classical straight edge and compass. (Via Kepler's triangle) so even if werzel = Reiku, the thread is not all a waste of time to look at.

 

He's attempting to make his posts look superficially like the posts of people like Guest in this thread, where lots of technical words are used and which seems to those people who don't know the meaning of the technical words a very tough to follow discussion. Whether this is a cry for attention or he honestly believes he knows the meaning of the words is something only Reiku knows but either way the posts of his in this thread (and I'd wager others) are meaningless.

Quite why he thinks its worth the effort to make such superficially high level but actually vacuous posts I don't know. The 'joke' he tried to play with previous accounts is not on us, those who do know what a tangent bundle is or what anomaly, but him. He learns nothing other than how to 'talk the talk' in such a way to fool people as ignorant as him. Even someone with mediocre knowledge of science (ie that of an interested layman) sees through him. It's a complete waste of time for him and actually counter productive to his supposed wish to become an actual physicist.

Ah well, we all knew he was going nowhere. All except him.

 

I haven't bothered to read the drivel that you've been posting Reiku, but it seems that you're trying to apply Noether's theorem because you're claiming that the Gaussian primes have some sort of symmetry?

Unfortunately you've missed the mark again, because, as guest says in the post that you have quoted "You notice that your Lagrangian is unchanged under the action of the continuous group," Your Gaussian and Einsteinian integers are not a continuous group so Noether's theorem does not apply.

 

Prom, he isn't trying to do anything, he's just posting spam in an attempt to get a rise out of us. It's obviously Reiku, by his ignorant postings and the fact he completely ignore my posts saying he's Reiku, then calling us morons, to try and get a response.

You're giving him WAY more credit than he deserves, since you are assuming he even knows what Noether's theorem is, all he's done is copy and paste Guest's post and put together a jumble of buzzwords.

Just report the thread and leave him be. He gets more wound up when none of us reply to him and he'll then get himself banned, if he hasn't already infringed enough rules to make that happen anyway.

 

Yes it is definitely Reiku (or his clone). Here is solid proof:

Either he is taking his medicine regularly or making some progress as now he can compose a grammatically correct sentence.

Reiku, when you join two sentences as in the first above, use ; not , between them. I.e. it should have been: "... forum anywhere; sciforums is a good cave ...."

 

If primes had a symmetry they'd be easier to find.

Thread closed. Please try to keep it civil in the future.