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Simple yet interesting.


Trurl
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Ok referring to https://en.wikipedia.org/wiki/Logarithmic_spiral the log spiral is given by [math]r = a e^{k \varphi}[/math] where [math]a > 0[/math] and [math]k \neq 0[/math] are real constants.

The sector area is [math]A = \frac{r(\varphi_2)^2 - r(\varphi_1)^2}{4 k}[/math] where [math]\varphi_1[/math] and [math]\varphi_2[/math] are your two angles.

So: What values of [math]a[/math] and [math]k[/math] have you chosen in order to get radial distances of 5 and 17; and what are the corresponding angles; and does the sector area work out to 85?

You should work this out and see what makes sense. You need to find the constants and the angles that make your picture valid.

And the primality of the radial lengths doesn't appear to be significant in any way, so can you explain why you think it is?

 

If there's anyone here who knows anything about spirals, I'm confused about something. In the picture on the Wiki page, they show the sector area as going between the origin and two points on the spiral. But clearly the radial lines pass over other parts of the spiral more near to the origin. What defines the sector, exactly? How do you know which arm of the spiral defines the sector? I don't know anything about spirals, I'm confused by this.

ps -- Oh I see what's going on. The radial line segment is defined from the origin to the point [math](r, \varphi)[/math]. So for example take a log spiral with [math]a = 1[/math] and [math]k = 1[/math], so the spiral is given by [math]r = e^\varphi[/math]. Then for an angle of [math]0[/math], the point on the spiral is [math](1,0)[/math] in polar coords. If you go around the circle again with [math]\varphi = 2 \pi[/math], the corresponding point is [math](e^{2 \pi}, 2 \pi) \approx (535, 0) [/math]. In other words the angle is always taken mod 2 pi, but the radius keeps growing (or shrinking towards zero as the angle decreases to minus infinity).

So if two angles are close together (within 2 pi of each other) the sector makes sense; but if not, it's unclear what the sector is. Anyway I'm beginning to get some feel for the log spiral. "Today I learned!"

Edited by wtf
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I got an angle between zero and 17 of 62.14599 degrees.

Is that what you got?

BTW, what are the errors of the 2sided triangle worked example I posted in the pictures. I wrote it 12 years ago. If it can be corrected, it would be useful. You should note it is inside a parabola. It is probably flawed but my description seems like a worthy effort.

I just gave you the angle now because I don’t have time to write out my work now.

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1 hour ago, Trurl said:

I got an angle between zero and 17 of 62.14599 degrees.

Is that what you got?

17 is supposed to be a radial length. What is 0 here? Can you clarify? What you wrote doesn't seem to make sense in the context of the problem.

Did you mean the angle between radial lengths of 5 and 17? Well if the spiral is [math]r = e^\varphi[/math] then the angles are [math]\log 5[/math] and [math]\log 17[/math], respectively, [math]\log[/math] meaning natural log. Is that what you're doing? 

From the sector area formula on Wiki, the sector area between radial segments of length log 5 and log 17 is (17^2 - 5^2)/4 = 264/4 = 66. That's with spiral constants a = 1 and k = 1. You could adjust k to make the area come out to 85, but I'm not sure what point is being made.

Have you looked at the Wiki page on the log spiral?

 

1 hour ago, Trurl said:

BTW, what are the errors of the 2sided triangle worked example I posted in the pictures. I wrote it 12 years ago. If it can be corrected, it would be useful. You should note it is inside a parabola. It is probably flawed but my description seems like a worthy effort.

Your log spiral changed into a parabola?

 

 

1 hour ago, Trurl said:

I just gave you the angle now because I don’t have time to write out my work now.

As Galois said as he was frantically writing up his mathematical discoveries the night before he was to die in a duel, "I have no time. I have no time."

Edited by wtf
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Quote

rom the sector area formula on Wiki, the sector area between radial segments of length log 5 and log 17 is (17^2 - 5^2)/4 = 264/4 = 66. That's with spiral constants a = 1 and k = 1. You could adjust k to make the area come out to 85, but I'm not sure what point is being made.

I mean from the origin to a radius of 17 the angle between is 62 degrees.

How did you square 17 and 5 without taking into account the angle between each. I think you missed a variable The angle is the unknown.

My answer may be wrong. I will write it up when I get a chance. But I also think you forgot the angles in the area equation.

Quote

Your log spiral changed into a parabola?

No, it ties in differently. I want you to also comment on that worked example but we are on log spiral now.

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The parable work was posted June 9th.

 

Quote

You could adjust k to make the area come out to 85, but I'm not sure what point is being made.

I propose there will be a pattern once we do this process with all possible SemiPrimes. If no pattern we can still manipulate the semiPrimes as we would a log spiral.

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14 minutes ago, Trurl said:

The parable work was posted June 9th.

 

I propose there will be a pattern once we do this process with all possible SemiPrimes. If no pattern we can still manipulate the semiPrimes as we would a log spiral.

The primality of 5 and 17 have nothing to do with anything, nor have you reached or stated any conclusion. You take any two angles and you will have radial segments with a sector area between them as computed by the formula on Wiki. It doesn't seem to mean anything. 

 

23 minutes ago, Trurl said:

I mean from the origin to a radius of 17 the angle between is 62 degrees.

 

What happened to the radial segment of length 5?

What point on the spiral corresponds to a radius of length 17? If the spiral constants are [math]a = 1, k = 1[/math], the spiral is given by [math]r = e^\varphi[/math]. What value of [math]\varphi[/math] gives a radius of 17? [math]\varphi = \log 17[/math], right? That's the angle, in radians. It comes out to around 161 degrees. [math]\log 17 \approx 2.8][/math], and 2.8 radians is around 160 degrees.

That's the angle it would make with the line from the origin to (1,0) if you plug in [math]\varphi = 0[/math].

How are you making your calculation?

 

 

Edited by wtf
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A=r(φ2)2−r(φ1)2 /4K

But you didn’t use the area equation right. Radius * angle squared - radius * angle squared divided by 4K

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13 minutes ago, Trurl said:

But you didn’t use the area equation right. Radius * angle squared - radius * angle squared divided by 4K

That's exactly what I did earlier when I computed the sector area between radial segments of lengths 5 and 17.  That's when you asked me why I didn't account for the angle. It's because the formula doesn't involve the angle, because given the radial lengths, you can compute the angle. 

That's this post:

https://www.scienceforums.net/topic/124453-simple-yet-interesting/?do=findComment&comment=1178970

Oh I see what you mean. I interpreted [math]r(\varphi)^2[/math] as functional notation. [math]r(\varphi)^2[/math] is the square of the radius associated with the angle. I didn't take it as multiplication. It doesn't make any sense to square an angle. I see your point, the notation's a little ambiguous but I think they mean to input the angle to obtain the radius, then square that radius. I could be wrong. 

Either way it doesn't matter. There's nothing about primes or semiprimes about any of this. You pick two angles and get a pair of radial lengths and compute the sector area, but it doesn't mean anything with regard to primes or semiprimes.

Edited by wtf
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16 hours ago, wtf said:

Either way it doesn't matter. There's nothing about primes or semiprimes about any of this. You pick two angles and get a pair of radial lengths and compute the sector area, but it doesn't mean anything with regard to primes or semiprimes.

Well that is completely unscientific. You did not prove it wrong. You took one value and came to the conclusion it has nothing to do with Semi-Primes even though you just took 2 factors and related the sector area to the Semi-Prime. So simple that it seems insignificant.

But the hypothesis exists to disprove. You have to creat an algorithm to graph all known Semi-Primes or at least enough to sample the data. If you believe that any of the 4 equations I posted show any pattern in the factoring of Semi-Primes, why would you think this geometric model would not have any truth?

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19 minutes ago, Trurl said:

Well that is completely unscientific. You did not prove it wrong. You took one value and came to the conclusion it has nothing to do with Semi-Primes even though you just took 2 factors and related the sector area to the Semi-Prime. So simple that it seems insignificant.

But the hypothesis exists to disprove. You have to creat an algorithm to graph all known Semi-Primes or at least enough to sample the data. If you believe that any of the 4 equations I posted show any pattern in the factoring of Semi-Primes, why would you think this geometric model would not have any truth?

It doesn't have any truth OR any falsity. You haven't stated any clear proposition that would have a definite truth value. You haven't said why radial lengths of 5 and 17 are meaningful. I could do the same exercise with lengths of 4 and 6. But I don't mean to argue with you. I learned a little about the log spiral and perhaps helped you to clarify some of your thinking, or not, as the case may be. 

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Clarity and argument are good. I post to solve problems and have good discussion. I envisioned the log spiral to increase in magnitude. The magnitude being the arc of Prime factors, 5 and 17. I propose as the magnitude increases so does the log spiral and it’s sector. That is given. But what if we graph 3*5 then graph 5*7 then 7*11 then 11*13? Will we find relations between their angles and areas of the sector. I don’t know of any attempts to do this.But other the circular functions and a spiral I don’t know where this idea could be tested geometrically

 

If you took 3*5 then 3*7 then 3*11 ... then 3*5 then 5*7 then 5*11... and graphed them on a log spiral at the very least it would organize the Primes. But what if you tried a number that wasn’t a Semi-Prime would the area match the product of the 2 products that form the radius? Does the log spiral formed by 5*17=85 fit all Semi-Primes?

 

Can we mathematically change the spiral so that we can test manipulations like a sine curve for alternating current? Can we find patterns in factors only seen in Semi-Primes:

 

(x^3/N) approximately= (x^2/y)

Here you could invert the fractions and see if the area between x and y, N is divisible by x^3 as y is divisible by x^2

 

There are other tests such as:

(25/17) approximately= (85/58)

 

That is my argument of the log spirals usefulness.

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57 minutes ago, Trurl said:

Does the log spiral formed by 5*17=85 fit all Semi-Primes?

You have not demonstrated this relationship for 5, 17, and 85. The best you can do is compute the sector area for 5 and 17 and then adjust the spiral constants after the fact. You'd then have to pick different constants for every pair of numbers.

And you could do exactly the same for 4, 6, 24; or 8,10,80. First, the primality of the first two numbers isn't needed; and secondly, for each pair of radial lengths you need to adjust your spiral constant after the fact. Don't you see this? 

What are the spiral constants for 5 and 17?

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14 hours ago, Trurl said:

That is my argument of the log spirals usefulness.

This popped up on my radar this morning. "Finding primes using a parabola." Looks like it might be of interest to you.

https://www.cantorsparadise.com/finding-primes-using-a-parabola-a939c6e8be53

First time I clicked on it the story appeared, then after that it said I needed a paid app. Not sure how this works but perhaps you'll be able to view it.

Edited by wtf
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Awesome link. I wonder what they used to create the graphics.

 

That is different from my attempts. They just have a sieve. I am trying to find a log spiral where the area equals the Prime products. The area should not equal with numbers that aren’t Prime.

 

The parabola I created isn’t a sieve. That is why I asked you to look at the worked example. I tried to write a mathematical proof.

 

I enjoyed the link. I have never seen other people relating Primes to parabolas. It seems like just a sieve and it is very complex. I know my looks complex like any factors will work with any number but it should only work with Prime products.

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On 3/3/2021 at 7:15 PM, Trurl said:

Remember I am claiming I can factor semi-Primes and thus RSA cryptography would be no more.

Isn't the semi-primes in RSA based on two random prime numbers?

On 2/25/2021 at 2:38 AM, Trurl said:

Is there a pattern?

I’m curious about a fundamental aspect of the ideas you describe; how could there possibly be a pattern in these conditions? Remember that the starting point is unknown numbers. Nothing is known about the numbers except for one common property; they are primes. You multiply* two random primes and expect a useful pattern** to form, how? What is the basics for that? What algebraic or geometric (or other) mathematical properties could possibly result in a pattern since you start from random numbers? I do not claim to have an answer, I'm just curious how there could possibly be any patten. To me the idea sounds like trying to use a series of coin tosses to predict the future outcome where tossing a fair coin. You could of course predict the average given a large enough set of samples, just as you could predict the average value for n in RSA given a large set. But again, my analogy could be incorrect due to misunderstanding on my part.

 


*) I know about product distribution, constructed as the distribution of the product of random variables, but that does not seem to be related to the ideas presented in this thread.
**) A pattern that allows the two factors to be calculated much more easily than using currently known or established methods. This is my interpretation of the goal and motivation from the descriptions posted so far.

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Yes a semi-Prime based on two random Primes. I refer you to the four equations already in this thread. Plug and chug. I know plug and chug is referred to as improper math, but here we are just testing values. I know I can’t yet find x perfectly. But the equations show the pattern between the semi-Prime and the smaller factor x.

 

I know you are think so what if we know PNP and x we know y. Point taken. But you can test for x by graphing the equation. Which I am arguing is faster then recursive division.

 

The pattern I am trying to draw on the log spiral is from the four equations. The equations do not hold true for non-semi-Primes. So I figure if I can come up with a geometric model of the equations a pattern may result in the model.

 

I need to draw the log spiral, I know. My hypothesis was that if I could relate the area of the sector to the semi-Prime at arc lengths of the Prime products a pattern would result. That is because the equations are only true for Prime products.

 

I know, I should draw the log spiral and prove it. But to do so is very labor intensive, so I though I get feedback on the forum.

 

The link WTF posted had beautiful graphics but it was just a sieve. I don’t know how to program my log spiral. It should be possible but it is very complex.

 

Hopes this helps explain my reasoning. Plug and chug the equations. That should show the pattern in factors. A pattern unique to semi-Primes. It really isn’t a pattern of Primes put a pattern in factoring. Like in science where you can’t find answers directly, instead you use what’s known and find the pattern indirectly.

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There is a chance my spiral may “break the rules” of a log spiral. If so it doesn’t matter it would just be a custom spiral.

I need to draw it.

From here on out I will call this project the Pappy Craylar Conjecture.

 

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  • 2 weeks later...
Posted (edited)

 

On 6/20/2021 at 10:50 PM, Trurl said:

Plug and chug the equations. That should show the pattern in factors. A pattern unique to semi-Primes.

Can you use the pattern to distinguish semi-primes from non semi-primes

Also I do not think you answered my question; how could there be a pattern? 

I'll try to add an analogy to illustrate the question further. Below is a scatter plot of two classes of random integer pairs. If both integers (x and y) in the pair are primes then the point (x,y) gets a color indirectly represent semi-primes. if x or y isn't prime then the point (x,y) gets another color. I see no pattern; I can't tell which color, red or blue, that actually represents semi-primes. I also can't predict which color the black dot should have; I can't tell if the dot represents a semi prime or not. In my example it is easy to create the plot because I know x and y. But there is not enough information to classify an unknown (x,y) pair or draw conclusions about semi-primes.plot.png.fae168c8b58d1cd10ce3a3cc484beaba.png

 

Note: I do not claim that a pattern can't possibly exist but if the random generator is fair, then what kind of algoritm could predict which class the random black dot belongs to, red or blue?

Edited by Ghideon
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Sorry for the delay. I was on a trip.

Well my goal is to convince you a pattern exists.

The scatter plot does not show a pattern, but that is true of many patterns. Although some one YouTube did show that Prime numbers in the coordinate plane fits a logarithmic spiral. It is different from my work but I say the 4 equations show the pattern. If you graph the equation for one Prime number and a domain of possibly x values, it is an inverted normal distribution.

The pattern is in the equations. Without that pattern we have no pattern. So if you do not see a pattern in the equations there is no pattern.

The log spiral is separate from the equations. It uses areas. I am working on the drawing. No guarantee it will work, but I just want to show my logic.

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9 hours ago, Trurl said:

Well my goal is to convince you a pattern exists.

The scatter plot does not show a pattern, but that is true of many patterns. Although some one YouTube did show that Prime numbers in the coordinate plane fits a logarithmic spiral. It is different from my work but I say the 4 equations show the pattern. If you graph the equation for one Prime number and a domain of possibly x values, it is an inverted normal distribution.

The pattern is in the equations. Without that pattern we have no pattern. So if you do not see a pattern in the equations there is no pattern.

The log spiral is separate from the equations. It uses areas. I am working on the drawing. No guarantee it will work, but I just want to show my logic.

I know where you claim the pattern to be. I want to know how there could be a pattern.  

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“I know where you claim the pattern to be. I want to know how there could be a pattern.”

How could there not be a pattern? How is there a pattern in DNA with billions of proteins? It is just a model to better understand relationships. Do patterns exist or do you create them? Graphing the equation for possible x’s for given semi prime should show a pattern of x.

A scatter plot may or may not show a pattern. But most things that are described by equations is a first step in finding patterns, especially when they are graphed.

The graph of the logarithmic spiral is separate from the 4 equations. It is in polar coordinates. The logarithmic spiral could fail if the areas did not fit logarithmically with different Prime factors. But if you graphed the same Prime number with all possible Prime numbers a pattern may result:3*3 3*5 3*7 3*11

The reason I believe there is a pattern is from the log spiral model. Most attempts to find patterns in Primes is solving series. I used geometry and equations.

I was a graphic artist. They teach that when you draw something it helps to break the scene into basic shapes. Obviously the world is chaotic but if you were to 3D model a person you would break them into polygons.

And yes, I believe there are patterns in everything. If there weren’t we would not have mathematicians.

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25 minutes ago, Trurl said:

How could there not be a pattern?

Ok.What is a pattern to you? What is the benefit of a pattern in this claim:

On 3/3/2021 at 7:15 PM, Trurl said:

Remember I am claiming I can factor semi-Primes and thus RSA cryptography would be no more.

 

34 minutes ago, Trurl said:

I was a graphic artist. They teach that when you draw something it helps to break the scene into basic shapes. Obviously the world is chaotic but if you were to 3D model a person you would break them into polygons.

 

Yes. And my understanding of the is that you try the opposite; predicting the person from too few polygons. Crude example: There is a pattern, all my models has one polygon. But that is not enough to find the unknown person below.

image.thumb.png.ffc9765a43856f5969dadb6767b4862f.png

I would like some explanation of, or reference to, the mathematical principles that you believe could predict the factors of a large semi prime when the factors are random. 

 

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“Remember I am claiming I can factor semi-Primes and thus RSA cryptography would be no more.”

 

This comment is to create interest. It is why we are finding semiprime factors in the first place. If I could factor semiprimes with the Pappy Craylar method it would break RSA cryptography. RSA is the first public key crypto. There is no longer prizes for cracking it hence the Pappy Craylar method is shared here.

 

 

“Yes. And my understanding of the is that you try the opposite; predicting the person from too few polygons. Crude example: There is a pattern, all my models has one polygon. But that is not enough to find the unknown person below.”

 

I should clarify that I am not finding the persons volume, I am referring to the person’s shape, the actual drawing.

 

 

 

 

“I would like some explanation of, or reference to, the mathematical principles that you believe could predict the factors of a large semi prime when the factors are random.”

 

 

 

 

 

 

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

 

 

x - Sqrt[ [ ((x^2 * pnp^4 + 2 * pnp^2 * x^5) + x^8) / pnp^4]] = 0

 

 

 

Graph for x=0 to pnp; where y = 0 is your x, where x > 0

 

There are no tricks. I welcome the questions. I think I have something. But you never know with math.

 

BTW if you can solve for x=0 without plotting it, it’s groundbreaking.

 

And the graph of what you just plotted shows a pattern.

 

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Thanks for the answer. Yes the RSA reference caught my interest. So far the only pattern I see is that however I try to formulate a request for further information about the fundamentals the ideas are based on it gives the same result, a reference to the four equations and that I should use them:

On 6/20/2021 at 10:50 PM, Trurl said:

I refer you to the four equations already in this thread. Plug and chug.

 

On 7/11/2021 at 12:59 AM, Trurl said:

I say the 4 equations show the pattern.

 

On 7/11/2021 at 12:59 AM, Trurl said:

The pattern is in the equations.

 

21 hours ago, Trurl said:

Graph for x=0 to pnp; where y = 0 is your x, where x > 0

My conclusion at this point is that no information exist in the mainstream science or mathematics that can help me gain enough insight in these ideas. At my current level of knowledge and interest the equations are of no interest and not useful. I may revisit this topic if new information becomes available.

Edited by Ghideon
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