# Simple yet interesting.

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11 hours ago, Trurl said:

Why did no one recommend Excel for the graph and the list? It would graph and list an N of 100 digits easily. It has a segment of adjustable accuracy.

Simply because I did not think of it, I did not understand your method enough to see Excel (or graphing software) as part of the solution. And excel number precision is 15 digits* and the method seemed to require more precision to be of practical use.

*) Source: The list of specifications at support.microsoft.com/

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I am only kidding. I did not think of it either. My nephew said he had a friend that did math on excel and mathematics. But it didn't click because they never taught that in school. I was watching a video of The Great Courses and the instructor showed how computers solve differential equations.

It is a cheap and powerful method. But it may be able to graph from 0 to pnp.

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• 4 weeks later...

Here is my previous attempt at a perfect equation to find semi-Primes. The trouble is that it is too complex to solve easily.

x = Sqrt[ [ ((x^2 * pnp^4 + 2 * pnp^2 * x^5) + x^8 / pnp^4]]

[pnp^4 = [[  ( (pnp^4 * x^2 + 2 * pnp^2 * x^5) )]   / x^2 ]  – [(x^3 / pnp / 2)]

Test these equations. If you want to know how they were derived pm me and I will send you a link to download free. I don’t mean to advertise, but if you are interested pm me. It is the easiest way to get you a lot of information.

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Posted (edited)

46 minutes ago, Trurl said:

The trouble is that it is too complex to solve easily.

One trouble is that the first one has unbalanced parentheses:

46 minutes ago, Trurl said:

x = Sqrt[ [ ((x^2 * pnp^4 + 2 * pnp^2 * x^5) + x^8 / pnp^4]]

Edited by Ghideon
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Good eye.

x = Sqrt[ [ ((x^2 * pnp^4 + 2 * pnp^2 * x^5) + x^8) / pnp^4]]

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Posted (edited)

If you are interested in these equations visit my status page.

i stink at writing for mathematics. So much for publishing a textbook. My education professors said I was writing musings for the internet. It may not seem like it but I have an undergraduate understanding of math. I enjoy reading Michio Kaku’s books. String theory is pure math. I love plugging numbers. Can anyone suggest a site with string theory math for the novice? From what I understand the math of modern physics is out of control.

Anyways enjoy the link in about me page. Can anyone recommend a book about writing for mathematics?

Edited by Trurl
Darn tablet erased my paragraph
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Ok I’m going to bounce an idea off you. If it is total wrong it doesn’t mean it isn’t important. I began my work on Semi-Primes years ago with a vision that a logarithmic spiral could show a pattern in Prime numbers. I have heard others say this, but they were talking a scatter plot while I tried to make my spirals characteristics match the properties of the Prime numbers. And of course, it is easy to say a logarithmic spiral will show a pattern, but finding the pattern is the challenge. I attempted several attempts without success.

The Hypothesis:

If you draw an angle between 2 lines of length of 2 Prime numbers the vector result is a length of the Semi-Prime perhaps a pattern will form in that the spiral that encompasses the end point of the vector result will form a logarithmic spiral. A spiral that can be described mathematically on both the coordinate plane and the equation of a logarithmic spiral.

If we know the angle between the 2 Prime numbers and the vector resultant we know all sides of the vector triangle. We also know the slope of the vector resultant and its vector magnitude. The slope or derivative can be used to predict future values.

There is a problem determining if a number is Prime or a Semi-Prime. The 4 equations I listed on this thread should only work if both x and y are Prime and when x*y = pnp. Multiplying an unknown factor by a known Prime and placing in my equations  may become a way to test for Primality. If the equation produces correct factors we know both x and y are Prime numbers.

Also, there are infinitely many Prime numbers so there are infinitely many corresponding factors for x. For example 3 *5 or 3 *7 or 3 *11. For every Prime factor there is a Prime that can be multiple to produce a Semi-Prime. Multiplying 3 by a larger Prime factor  will produce a Semi-Prime that will increase in magnitude.

**Just to be clear on the triangle I am drawing.

I envision a vector with sides of the Prime factors. The angle between them will determined by SSS from the Prime number factors and pnp as the vector resultant. I propose that if we graph this for all known Semi-Primes a pattern (a useful pattern will result.)

Again, I am aware that this is easier said than done. I cannot calculate this by myself. But I propose that treating Semi-Primes in vector form will lead to better mathematical representation.

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Ok, this post has many views no comments on my new hypothesis. I do understand that it may be that no one is exactly sure what it is I am doing. I am not doing one thing in this post. The goal was to present my equations that solve for x knowing only pnp. I think we can agree that the equations are significant, but to find a practical solution we need the ability to solve large numbers. What works for 5 and 17 is easy to solve, but advanced computer programming is needed to crunch a 100-digit number.

I am trying to improve my ability of explaining mathematical subjects. I just finished reading the book Accidental Genus by Mark Levy. It dealt with producing content. I got content but I need to communicate my ideas. I believe explaining math is a skill that improves with practice. But sometimes a math solution is hard to explain or prove why it works and how we got there. And it does not help that most U.S. citizens are math illiterate. So, you show your work to those who do and prepare for criticism. I wrote “this problem can be solved with elementary mathematics,” in a lesson outline. And my instructor wrote back “this isn’t elementary mathematics.” I meant math below calculus. It was only one phrase out of a 5-page document. My material did not rely on this phrase. But it was enough to make her very mad and pick apart my lesson plan.

So that is where I am at. I’m sure if I had a person-to-person conversation with someone, I could explain my work. Using diagrams and paragraphs by themselves is a challenge. But I did not write this post just to babble. I want you to look at the attached pdf and see if it has any merit. In it I claim that I can solve a triangle knowing only 2 sides. Yes, with that title no one will bother to look at it. But spoiler alert I draw the triangle that with only 2 sides known with the characteristics I want it to have. Yes, I know only 2 sides has infinitely possibilities. But does it? What if we add more information? Perhaps the hypothesis I just posted. For example, what if you have a side of the triangle that is a product of the 2 sides? I know that sentence makes sense to me and not you, but what if we design a set of custom triangles? So that is what I meant. It still may not work, but I am working on a geometric drawing that may tie the 2 together. I am writing it up now. But it is important that I accurately communicate my idea.

Simply put this is it:

I believe there are vector “factors”, 2 sides of a triangle that through vector summation result in the pnp side (the known vector resultant). So, if we can find a way to relate vector sum to the factors of the scalar of resultant, we can factor pnp into 2 semi-Prime products. And since pnp is semi-Prime only one set of products were work.

**This does not mean that a semi-Prime vector resultant would not have infinite many solutions. Instead, the vector sum with the Prime products, would be unique. And since it is unique, we may be able to predict or design where it occurs.

**Again, this hypothesis probably makes sense to me and not you, the reader. But what does it make you think? It is hard for me to describe something where no source to reference. But any feedback is helpful. The goal is not to produce a theory, but to explain to you, the reader my reasoning.

If you are interested of all in my hypothesis, review the attached PDF. I will later attempt to explain why I brought up this subject again.

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Posted (edited)
6 hours ago, Trurl said:

I claim that I can solve a triangle knowing only 2 sides.

This statement is false on its face. That's why readers are put off and have no idea what you're talking about.

Could you give a specific example? Say I have a line segment of length 2 and another of length 3, both with one end at the origin of the x-y plane. Can you show how you get the third side? What additional information is supplied? Please be specific. No need for extraneous discussions of the travails of your long-suffering teacher. Just show an example of two lengths and how you determine the third one.

Also, what does pnp stand for?

Give a specific, completely worked out example and leave out all other extraneous commentary, and that will go a long way toward clear mathematical communication.

Of course if you have two sides of a triangle plus some as yet to be specified information, the additional information might be sufficient to work out the third side. Why don't you give an example so people can see what you are talking about?

Edited by wtf
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2 hours ago, wtf said:

Of course if you have two sides of a triangle plus some as yet to be specified information, the additional information might be sufficient to work out the third side. Why don't you give an example so people can see what you are talking about?

9 hours ago, Trurl said:

I am trying to improve my ability of explaining mathematical subjects. I just finished reading the book Accidental Genus by Mark Levy. It dealt with producing content. I got content but I need to communicate my ideas. I believe explaining math is a skill that improves with practice. But sometimes a math solution is hard to explain or prove why it works and how we got there.

Every responder so far has failed to understand where your mind is wandering to.

Dr Mark Levy is a medical doctor, I don't know the book but it doesn't seem to have done the trick; spelling is also important when communicating.
I know this because I have posted some awful howlers here due to bad spelling.

I picked this out because there is another Mark Levi, who is a Mathematician and the author of this book that might interest and help you.

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Thanks Studious. I am going to get that book. I think it will help me communicate ideas. No one knows what I am saying. The ideas might just be too abstract. I thought the 4 equations were concrete. But you are right I have to come up with something tangible. One clear example. I included a PDF in the last post you read. Is that any help?

I do have the second part of my write-up. I am aware the following hypothesis most likely don’t work, but do you get any ideas when you read them? I am describing geometry. Easy enough to make a hypothesis, but I have not been successful. When you read them, do they make any sense?

Other examples of my hypothesizes. I want to post all the content. I will not post again until I have a concrete example. The attached PDF of my other hypothesis shows a parabola and an example of 64 degrees by a triangle with sides 3 and 5. If the right eyes see it, I believe it could be turned into something powerful. So please excuse these hypothesizes. It is related, but I can't explain it.

P.S. The reason I originally called it the 2 Sided Triangle was to spark interest. You are right people couldn't get over the name.

Also pnp is what it has always been in this post. PNP is the given semi-Prime. I am placing into a triangle pnp and its factors x and y. It would be extremely useful. But it relies on the fact that we can use the equations I derived to find x. Which we can if pnp is small. Try the 4 equations for known semi-Primes. My hypothesizes are what you can do once x is found. I also hypothesize if all sides of the triangle equaled pnp, x, and y. It is possible to draw such a triangle. There are infinite triangles, but I want to find what angle between x and y will produce side of pnp.

This post is already too long. I will post when I have a clear example. Or to answer questions. All this ties in. And maybe if I explain it right I can make you believe.

The following is a post I was working on before I read the replies. So something repeat.

Ok here is a short but powerful idea. I think we should write as many hypotheses as we can. A free writing exercise like Mark Levy’s book. A brainstorming session that is serious and not silly. Of course, some of you may find my hypothesis silly. In Stanislaw Lem’s book Futurological Congress it briefly mentions the futurists who study what words will be used in the future to determine the technology the future has. But this hypothesis is simple, but I know my drawings stink. If anyone knows a program to create traditional drafting and geometry please share. Adobe Creative Cloud is a scam.

Ok here is my hypothesis:

You see a line in the first figure of the attached drawing. This line is length pnp. There is a half-circle below it.

In drawing 2 there are three angles in the half-circles. These are the triangles whose largest side is pnp. They are many, but the angles must touch both ends of pnp.

The third drawing is an isosceles triangle. (The isosceles triangle is half white and half green.) It is drawn in this way because we only have test triangles (angles) who fall in the green area. These angles still have to touch both sides of pnp, but we only test those areas in green because of redundancy. If one segment of the triangle is large the other is short the lengths will repeat after half the distance has been covered. (I hope this makes sense. I need help with the development of this idea.)

That is it.

Just picture only using some of the circle and relate it to finding angles on the parabola of the previous post hypothesis.

Abstract hypothesis (Just for fun.):

The Riemann Hypothesis to prove or disprove there is a zero we can use a triangle who has factors as its segments. I believe ½ is a critical line where the zeros occur because the factors are on an isosceles triangle.

The sides are equal and cancel each other. Dealing with fractions so we cannot have division by zero.

I do not know enough about the Riemann Hypothesis to describe further. But this is just a brainstorm I had. I know its silly, but I just wanted to share the idea. See attached picture.

When you use the 4 equations in this thread to find x from pnp, x occurs closer to the left-hand side of the number line. My future concentration will be on determining x perfectly. Maybe if we apply only the green area on the circle to searching for x possibilities, we might will solve semi-Primes with geometry.

I am aware my hypothesizes seem absurd. But that is how I formed the 4 equations I posted on this thread. I think there should be a math class on nothing but forming hypothesizes. I do not mean the hypothesis shouldn’t be well thought out. It may even be to an impossible problem. But I just read Mark Levy’s book on free writing and I want to do math free writing. But often choosing a math problem to work on is just as important as solving it.

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Studiot, when you  recommended Mark Levi’s book did you see a parallel in that it was as confusing as my posts? I mean the guy is a mathematician and physicist so he is more skilled than me. The majority of reviews are 5 stars. But the text was not user friendly. It is...proof...diagram...proof with one or two sentences to explain. It would take me hours each proof to find out what he is talking about. And he said you only need geometry. I could take his word the proofs work, but wouldn’t that defeat the purpose?

I loved the first chapter where he was going to relate pure math to physics. But I couldn’t make it past the second chapter. I looked ahead and I have no clue what direction he is comparing math and physics.

If my posts are explained this poorly (or worse), I have to fix it. I need to define my ideas like a textbook would. Like Levi’s book, he may have the best relation of practical science and pure mathematics, but if it takes months to recreate his work, no wonder my ideas are not clear.

I am working on a clear write-up. There is so much information it will take weeks.

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10 hours ago, Trurl said:

Studiot, when you  recommended Mark Levi’s book did you see a parallel in that it was as confusing as my posts?

Thank you for coming back to discuss.

No the parallel was definitely not that I though your posts confusing.
Actually I did not find what you said confusing, just that it did not seem to lead me anywhere.
To explain my sugestion about the book. I reckon he was thinking 'ouside the box' and you are perhaps doing something similar but having trouble crystallising your ideas and setting them down in a presentable order.
They may or may not be good ideas but until you can express them understandably to others they will remain undisclosed.
You should not think of Mark Levi actually proving Mathemtics in a way acceptable for a formal maths text or treatise.
But that he can demomonstrate processes in the natural world that follow them helps some people and shows there is a connection beyond mere symbolism. And he has some very interesting (to me anyway) ideas on this. I think he overdoes Pythagoras, but demonstrating the Gauss-Bonnet Theorem by Physics is quite something.

Out of interest I find Stanislaw Lem very confusing. My wife and I once cooperated in a translation from Hungarian to English of some of his works.

Personally I am not very interested in number theory but I was intrigued to find out more about this emboldened claim of yours

On 5/29/2021 at 1:00 AM, Trurl said:

So that is where I am at. I’m sure if I had a person-to-person conversation with someone, I could explain my work. Using diagrams and paragraphs by themselves is a challenge. But I did not write this post just to babble. I want you to look at the attached pdf and see if it has any merit. In it I claim that I can solve a triangle knowing only 2 sides. Yes, with that title no one will bother to look at it. But spoiler alert I draw the triangle that with only 2 sides known with the characteristics I want it to have. Yes, I know only 2 sides has infinitely possibilities. But does it? What if we add more information? Perhaps the hypothesis I just posted. For example, what if you have a side of the triangle that is a product of the 2 sides? I know that sentence makes sense to me and not you, but what if we design a set of custom triangles? So that is what I meant. It still may not work, but I am working on a geometric drawing that may tie the 2 together. I am writing it up now. But it is important that I accurately communicate my idea.

I am also wondering if you are grappling with ideas of algebraic geometry ( & topology).

These were introduced when it was discovered that the basic ideas (elements) of point, line, plane and hyperplane do not fully cover or describe (since 'cover' is a special word in maths) even the extension to 2 dimensions let alone 3 or more. Some additional basic elements were required. And one type is based on a triangle.

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All my hypotheses in this thread rely on these 4 equations being true.

p3 = ((pnp^2 + x^3) / pnp) – ((pnp + (x^2 / (pnp^2 + x)) * pnp))

p5 = (pnp^2 + x^3) / pnp -  (x^3 / pnp)

Separate equations. pnp=x*y

x&y are Prime factors of semi-Prime pnp

Don’t simplify, graph in software.

x = Sqrt[ [ ((x^2 * pnp^4 + 2 * pnp^2 * x^5) + x^8) / pnp^4]]

[pnp^4 = [[  ( (pnp^4 * x^2 + 2 * pnp^2 * x^5) )]   / x^2 ]  – [(x^3 / pnp / 2)]

2 separate  equations. pnp = x*y

pnp is the known semi-Prime and x and y are the Prime factors.

Can plug in pnp and x. Can we find x knowing only pnp?

_______________________________-

Obviously if x and pnp are known there is no advantage. But if you plot x on a graph it should produce a faster result than factoring.

Also if the equation works you could multiply one unknown x and a known Prime y to get pnp. If pnp is a semi-Prime number then x is Prime also.

If it were possible to find x knowing only pnp, then it would mean I just solved a triangle knowing only side pnp. The other two sides would be x and y the Prime factors. I know you can’t solve a triangle knowing only one side, but I have chosen to graph pnp with sides x and y.

I am choosing that triangle. For example a pnp of 85 and and x of 5 and a y of 17 would give me a triangle with all three sides known. I know the angle between x and y would be a large obtuse angle. This angle changes as you graph 5 and 23. There are infinitely many 5 * y = pnp. I chose to draw this triangle just because it is a simple picture of what is happening with semi-Prime factors.

I don’t know if a pattern will result, but that is the geometry I’m using.

The reason I said 2 sided triangle was to create interest. Looking back, it made people not read my pdf. My drawing is simple. Does it work? I don’t know if it will prove useful. You can graph the equations to find x. Then you take a second graphs of all known factors themselves such as 5 * 7 or 5 * 11  or 5 * 17 or 5 * 19 or 5 * 23  and 5 * next Prime.

That explains all my work. I still have to show a worked example and draw diagrams.

I know you mentioned topology, Studiot. The x, y, and pnp are all in the 2D coordinate plane. The triangle I am solving is just trigonometry. SSS, find the 3 sides then solve for the angles. I don’t think there is anything wrong with that approach. However, you are questioning whether it is useful. That is why I posted it. Mark Levi was doing things with triangles. He explained why a fluid will form an equilateral triangle when  rotated. I was unaware of applying pure math to physics. So I guess I was doing something similar trying to apply vector addition to factors. Has it been done before? I am confident that my 4 equations can show the relationship between x, y, and pnp. However, I haven’t solved for x knowing only pnp. However I believe there is a pattern in these 4 equations that is significant. It may not be the triangles I am drawing, but my main hypothesis is that a pattern in semi-Primes may signify a pattern in Prime numbers.

All unproven. But that is why I posted it.

My question to you is: can vectors be applied to factors? I believe they can. But my triangle means nothing without the 4 equations. I had an abstract thought that I cannot explain. But I think the drawing I posted from the Numberphile video has a critical zone of ½ because the factors are acting as vectors. I know that is easy to say but I can’t represent it mathematically. But I have a sense that factors are acting in an isosceles triangle and cancelling each other out and resulting in zero. Laugh at me if you want. But if you could express the factors of a number in terms of a vector, patterns would result.

BTW, I’m not sure how Lem’s books translate. There are a lot of adverbs and adjectives. But The Cyberaid is a good read for the sciences. Memoirs from a BathTub makes no sense.

You obviously have a more extensive math background than me. So I leave you with this question:

What is the geometric representation of a set of factors? If I gave you a compass and straightedge could you graphically represent factors. Not necessarily a number line, but fit the factors and their products onto a geometric shape.

I know the above may sound impossible but did not Mark Levi attempt it? But I think it is a part of mathematics that’s missing. Look at the circular functions and how they describe waves from electricity to springs. I think that is the true reason you recommended Levi’s book.

Can you relate geometry to factors?

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Posted (edited)
21 hours ago, Trurl said:

Can you relate geometry to factors?

I would still like to see a simple, complete worked example with lengths 2 and 3. Put the 3-side along the positive x-axis with one end at the origin; and show me exactly how you place the 2-side such that the angle is somehow uniquely determined by ... something.

Edited by wtf
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I am working on a example. You are right the idea may be too abstract to be useful. I was asking you guys if there was some relation to relate geometry to factors. I believe it may be possible. It doesn't have to be overly complex. I read once that we use the coordinate plane as our main reference for a coordinate system, but we forget anything geometric can create a simple, custom coordinate system. I want to make vectors relate to factors. I know that 5 * 17 = 85 would need an angle greater than 180. But what if the magnitudes of the 5 and 17 results in the same angles as 3 and 13 only the magnitudes are different. What if you had a vector of 5 and a vector 17 and the cross product equaled 85?

That is why I tried to use a logarithmic spiral as a coordinate system and have the angles equal and the magnitudes increase.

None of this is figured out. That is why I posted the hypothesis. But I do need a concrete example. I am just asking if anyone has any thoughts on a custom coordinate system or vectors related to factors. Or any geometry explaining factors.

Yes I know I need to know 3 sides to determine a triangle. But in pnp=x*y you take one known and factor it getting 2 unknowns. Well actually we are using trial and error. But if you could solve any one of the 4 equations, you have 2 unknowns from 1 known. The reason for the geometry is to find a pattern. You are correct in that I don't know this pattern. But that is why I posted. Studiot knew what I was trying to do (to relate geometry), but I have a feeling he thinks it is not possible. That could be.

But I will post again if I figure any system that incorporates factors

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40 minutes ago, Trurl said:

Studiot knew what I was trying to do (to relate geometry), but I have a feeling he thinks it is not possible. That could be.

Actually I have come across something connected to agebraic geometry and group theory.

I have just been trying to remember it.

But not to vectors. You require a whole lot of extra mathematical structure for vectors. I certainly think that is the wrong tree to bark up.

Look at it like this

15 =  5 x 3 ie it factorises into 5 and 3.

But 15, 5 and 3 are all numbers (integers to boot).

That is they are all the same kind of (mathematical) object from the same set.

This is a consequence of and consistent with the axiom of multiplication that for every a, b in the set a x b = c is also in the set.

However this is not generally true for vectors as it would require the product of two vectors to be a vector in the same set.
In you case you have talked of the vectors ' 5 and 17 in the plane so the product (whatever it is) must also be a vector in the same plane, which the vector cross product does not give you. Neither does the vector dot product.

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1 hour ago, Trurl said:

I know that 5 * 17 = 85 would need an angle greater than 180.

Why is that? Isn't there a right triangle with sides $5, 17$, $\sqrt{5^2 + 17^2}$  ?

Aren't there lots of other triangles with those two sides? The angle between those two sides can be absolutely anything strictly between 0 and 180 degrees.

Can you try to explain what constraints you are trying to put on the angle, and what they are based on?

Remember, coordinate systems are not inherent in geometry. The geometry comes first, and then you impose a coordinate system. The coordinate system can never change the underlying geometry, it can only make the equations simpler or more convenient in some cases. But the shapes, the angles and lengths, don't change.

20 minutes ago, studiot said:

However this is not generally true for vectors as it would require the product of two vectors to be a vector in the same set.
In you case you have talked of the vectors ' 5 and 17 in the plane so the product (whatever it is) must also be a vector in the same plane, which the vector cross product does not give you.

You could interpret plane vectors as complex numbers and divide them that way. I'm not sure what this line of thinking would do for the OP, who is having a hard time articulating his idea.

I don't think anyone would ever take the number 31, say, and note that "Hey that factors as the dot product of (2,3) and (5,7)." Pretty wild thought ... Wonder if it means anything. Given any integer we can form the set of all the pairs of vectors whose dot product is that integer. This can't possibly amount to anything, else we'd have heard of it, especially in physics in the context of Hilbert space inner products. There must be some geometry in here somewhere ...

Edited by wtf
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Ok. You wanted a worked example.

This is for “crafting” a triangle. There is a high probability it doesn’t work. But I wrote it in 2009 and am trying to relearn my reasoning.

This isn’t the “vector factors” I talked about. But it is a geometric representation of algebra. It is what is in the pdf I posted but you probably haven’t seen it. It is based in a parabola. I just want to know if it works. No one has ever tested it. I know it might appear absurd but I was inspired when I wrote it.

I will address the current discussion in a future post. It is difficult to relate vectors to factors. But I still think it be useful to create such a model.

Refer to the images for crafting triangles.

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Back to the discussion.

I concede using triangles to relate factors to products is challenging.

This is the best I could come up with:

For semi-Primes and their product where  5 * 17 = 85 a logarithmic spiral with radius 5 and 17 with the angle between 5 and 17 equal to the angle where the area of the logarithmic spiral encompasses an area of 85.

You just design the spiral using the known 5, 17, and 85.

Do you see any advantage to doing this? I am working on a worked example. The reasons I think it is useful is that it is simple, includes all 3 values, is reproducible, and properties of log spirals are already known.

I envision a logarithmic spiral that has semi-Primes positioned along similar angles that occur at different magnitudes. That is similar angles reaching to different lengths of the logarithmic spiral.

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On 6/11/2021 at 4:54 PM, Trurl said:

For semi-Primes and their product where  5 * 17 = 85 a logarithmic spiral with radius 5 and 17 with the angle between 5 and 17 equal to the angle where the area of the logarithmic spiral encompasses an area of 85.

The log spiral has many interesting properties. For example, "the distances between the turnings of a logarithmic spiral increase in geometric progression." Perhaps this relates to something you're interested in.

Can you draw a sketch of how 5, 17, and 85 relate to the log spiral? I think this might make your idea more clear.

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I still need to test using equation, but a log spiral with a radius from 5 to 17 and a sector area of 85. Just have to plug and chug into equation to be sure an area of 85 can be reached

Excuse the graphic. It is rugged but shows what I mean. I am a trained graphic artist but I face compatibility issues since most programs are subscription based. I have to change the way I design and what tools I use.

If this works I will draw a triangle.

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