Trurl

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About Trurl

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    applied mathematics
  1. Prime Products just one last time

    I know this doesn’t solve the problem. I am only trying to solve the triangle I constructed. But is it possible to find FE by subtraction. I may not be doing it right, but is it possible to do subtraction of the triangle segments to get FE? To me it seems possible, but when you go to do it, it is confusing. But it just feels possible. Remember I am only trying to solve for FE. AC = N [Absolute value [ AC – AE – (AC-CE) – CE]] = FE
  2. New TSP Method (p=np)

    I will stay out of the conversation after this post. I just wanted to clear up what I said. It may make more sense than you think. All I am saying is that geometric constructions may simplify the calculation. Let me know if this does or does not make sense. I am saying take a compass and draw a circle that encompass 2 points. Then with 3 points draw another circle from the 2 points closest together. Keep drawing circles from all points. When there are several points, the circles should intersect somewhere along the circle. Connect those intersections with lines and you have a polygon or the shortest path. Do you remember in high school when they taught us to find the center of a line by taking a radius more than half of the line and string 2 arcs from each side? The line between those arcs passes through the center of the original line. I propose if you did the same constructors of circles on the unknown points, geometric constructions such as finding the tangent of a circle (or any of the dozens of circle constructions) you would simplify the computer process. I don’t know how to put a drawing compass into a computer program, but you could always use a CAD script. But then again, I don’t know the algorithm to such a thing, as I have only spent 2 minutes coming up with a hypothesis. I would say that a radius of the circles, and arcs of those intersecting circles, would eliminate calculations that just can’t be done. I wouldn’t scrap the polygon idea. Instead, I’d use the polygon to form a path between the intersecting arcs. Also by using circles you have the advantage of all the circular functions. Do they still teach geometric constructions in geometry? Yes, I know the problem has N points, but this is a simple approach. As more and more points are formed you would have to erase (delete) some the circles no longer needed. And no, I don’t claim I can solve this problem. But I do like how the computer was run to solve this problem. Computers have already ruined chess and that game with squares. I’m glad there is much work to be done to solve this problem. I hope this is clearer. All I am saying is use geometric circle constructions. After all, you can build almost any shape with them. Someone has probably tried it, but before modern programming, such an idea wasn’t possible. Because in CAD you could program it to draw hundreds of points.
  3. New TSP Method (p=np)

    Polygons seem so complex. I like this idea, but circles are more closely related to navigation. Here is my humble opinion (not understanding the complete traveling salesman problem): Draw circles at a start point. It doesn’t matter which one. Continue to connect points with circles. Like at technical drawing or highway design where circles intersect is the path. So you would 2 circles between 2 points. Then you would draw a circle from the 2nd point to the third and where the intersection occurs is the shortest distance. The trick is to program it on the computer, taking away circles that no longer give the shortest path. I believe this is what you did with the polygon that encircles all points. However, I believe it is overly complicating your method. Brilliant none the less. Will you post your computer program here to share? I’m not concerned it is similar to other approaches. It is your approach. I suggest you get a technical drawing book and look at the geometric constructions. Circles not polygons, because you have your polygon acting as a circle. Circles are much more suited for the task.
  4. Intelligence test

    Really? This is an incredibly unreliable way to test for IQ. How does an intelligent person spend their time? They read books and study all day? How does an intelligent person interact with others? This implies that every person of a given IQ responds exactly the same way a different person of the same IQ acts. That is a ridiculous claim. If you think it is good science to only to test one aspect. That is why if a psychologist gets a test that someone excelled or did not excel, they evaluate further. One test might show a high I.Q. but isn’t more of an intelligence test if you actually know something about the person? If someone said a pilot should have an I.Q. of 120 or above, would that mean everyone above 120 should fly a plane? I have read studies on creativity. Specifically, Mihaly Csikszentmihalyi. He did not go around passing out I.Q. test. Instead he rigorously interviewed creative people and looked to see what they had in common and how they differed from others. That is even worse advice. He is no professional (neither are you) and even if you were, it would still be bad advice. IQ tests are done by several people who have studied this and know what they are doing. Even they may not output a reliable test. And don't encourage him to do even more unscientific hypothesizing. I'm sure if he came up with an IQ test, it would completely confirm his hypothesis in this thread. Why is designing your own experiment bad advice? The APA format is designed to introduce an experiment and show the results and findings. If you cannot find a I.Q. test (which is hard for the armchair psychologist), what is wrong with designing your own based on research, to fit the needs of the study? I think it would be a lot more fun and be more meaningful then passing out some test you don’t understand let alone did not designed and not have it meet your needs. Firstly, ''scientist'' is a broad term. You can't challenge a mathematician or a chemist to design an IQ test based on the fact that he is a scientist. Secondly, ''why don't you do it?!'' isn't a valid response in proving or disproving a theory. It's like when you criticize a movie and call it bad and someone says ''why don't you make a better movie??!?''. That's not how this works That is not intended as an insult. I truly would like to see this community work on such a project. With the knowledge of this community surely some here is a practicing psychologist or has been through I.Q. testing. I am not saying make a better movie, but have write one of those YouTube alternative endings.
  5. Intelligence test

    IMHO, I think you should design your own I.Q. test. I don’t mean start from scratch, but make a hybrid test that tests the skills you are looking for. Searching for an I.Q. test on the web will lead to a bunch of garbage or some high-priced package. I don’t know what your goal is. However, usually psychologist use many characteristics to determine I.Q. Testing is just one. For instance, they can look at grades in class, or how a person interacts with others, or how they spend their time. That is why I don’t think you are going to find a credible test that allows you to test people over the web. Of course, I’m just an armchair psychologist. But I believe I am giving you good advice to make your own test. After all, you are the only one how knows what factors you are looking for. Here is some credible tests, which may or may not relate to I.Q. However, those who created them knew what they were looking for in the results of the test. · SAT · ASVAB · AHSME - http://artofproblemsolving.com/wiki/index.php?title=AHSME_Problems_and_Solutions I hope this helps. It may not be what you are looking for. But without records of I.Q. of a sample group, you must be resourceful. Note that the AHSME test is very strategic. The rules are listed on the website with previous test. It is a timed test. And you can learn a lot about the student from the problems they choose to complete, because the student losses point for wrong answers and must break 100 points. There is both knowledge and strategy involved. Also, the problems are not grouped by subject and the word problems are not just rearranging values. There is problem solving. That mean no memorizing answers or not working through the problem. In fact, I propose a challenge to the members of SFN and the Moderators. If you are the scientists you think you are: I Challenge the members of SFN to design their own I.Q. test. If you think that Chriss is not knowledgeable enough to test the I.Q. with analytics then post a question to this post that you would consider a measure of I.Q. until the community has designed a credible solution.
  6. Obtaining IQ test database

    I have read all this post. And I am not going to criticize you for looking for patterns in numerology or astronomy. That doesn’t mean I correlate I.Q. with the stars. But I believe that numerology does have interesting patterns, though I have never really studied it. But if you knew when a person was born, could you predict when the next person of 160 I.Q. was born? You wouldn’t need to use stars. You could test your prediction in astrology, but IMHO, such a pattern would be hard to find because of the complexity of the data. Predicting when a person of genius I.Q. being born would be like predicting the weather. You would know that a chance of rain is likely, but have only an educated guess when it is going to happen. I’m an armchair scientist too. And I think it would be more practical to match I.Q. to genetics. But I believe there is more to Us than genetics. I would rather have my I.Q. determined by the stars than have someone say I am just a product of evolution. But that is my opinion and not science. For an I.Q. database, I would say that public school in the U.S. would have that data. When I was in high school and below we took a standardize test every 2 years. It was called the C.A.T. test. The school used it to place kids in the gifted program. But with today’s computer analytics, you could image that they probably have 40 years of test on thousands of students over the lifetime of their schooling. Another place for advanced data on I.Q. would be military records. Again, having access to such a database isn’t possible. (The same as public schools.) From one armchair scientist to another. I would email academics and ask them what they use for such research. I would try and prove that the I.Q. wasn’t random. I am not skilled in those mathematics, but there are mathematicians who believe nothing is random. I don’t think that means that humans can always see a pattern in data. I first think you should look at the patterns that already exist. If you did map the stars to I.Q. what set of rules would you use? Would you use the constellations like horoscopes are determined? I know if you are not an armchair scientist that such correlations don’t exist. But isn’t time based off the stars? I’m am not in agreement that stars determine I.Q., but you should find the data you need and find the answer for yourself.
  7. The Physics of Star Trek

    Here is my attempt at science fiction. It is a short story I wrote for a creative writing course over 7 years ago. I reread it. I didn't remember the story. It was like reading it for the first time. But be honest and let me know if it makes you think. You can be honest because I won't be offended. I am not a writer by trade. story013_final_webcopy.docx
  8. The Physics of Star Trek

    Excellent reply. I just was thinking that instead of anti-matter, anti-gravity would more efficiently propel ships. Instead of breaking up energy and using that massive amount of power released to power a warp engine, using the gravity that already exists in matter, would make more sense. We already sling-shot satellites off planets orbits to speed them up. If gravity was controlled there would be more versatile and realistic solutions. Also, if artificial gravity did exist, wouldn't it just cause forces that would terminate the inhabitants of the ship. When a rocket leaves for space the G-forces exerted on the astronauts are significant. So artificial gravity on the ship would mean that a warp drive would increase its forces. But again, if there were artificial gravity, it would be a better alternative to navigate the ship through space by using gravity. Of course, as you guys mentioned fictitious physics can't be proven. I'm just making this stuff up. But has anyone here ever thought to use their science knowledge to write fiction? I know it's been done before. But what about in college when you had to take English classes? I wrote a short story for class years ago. It didn't get its message across. I had a lot of symbolism in it, but unless you knew why I wrote it the way I did, the point was lost. I'm going to read it again and see if it is worth anything.
  9. Prime Products just one last time

    Attached are 2 important PDF’s that have been on my site for years, but are probably only seen by a few because of the difficulty of going through all the information. Yes, some patterns may repeat. But here in the first PDF, 3 equations are presented. Pay close attention to: Sqrt[(N*y – x^2)/x] = y This is one pattern for sure that I know isn’t just repetition. It is used in the equation at the beginning of this discussion here at SFN. The other 2 equations are just hypothesis. You will see this in the second PDF: I believe with most symmetric key ciphers, the fact that Prime numbers equal a one way function does not mean that there isn’t a pattern in the one way computation. So if N equals the product of 2 Prime numbers, Prime numbers have no pattern, but multiple 2 Prime numbers together there is a pattern. Where x = 571 and y = 1381 1381 / 2p = 219.7929764 2p * (remainder(1381/2p) * 2p) * 571 = 13208.65186 rewritten: 2p * ((1381/2p-whole number part) * 2p) * 571 = 13208.65186 13208.65186 / (1381/2p) = 60.09 13208.65186 / 60.09 = 219.7929764 219 = 219 And another relationship: New equations: 2p * (remainder (17 / 2p)) = 4.42 4.42 * 5 = 22.1 22.1 * (5 / (17* 2p)) = 0.884 0.884 * 5 = 4.42 4.42 = 4.42 See what you can do with it and share it. I believe p stands for Pi. Some of these write-ups are several years old and after making so many I don’t always know what I intended to communicate, without studying them. But I feel these PDF’s are important. That is important enough these write-ups may give meaning to the work. Either way, let me know if you think they are trash, or if it gives you any ideas. 20140519TrigPrimes005secCopyCC.pdf PrimeProductSolutionFlyerCopyCC.pdf
  10. The Physics of Star Trek

    I think “Space” should have its own thread here at SFN. I want to do a lot of reading on space. Such topics as: · Space Simulators · Space Dynamics · Terraforming · Mining Space · Space Station Design · Moon Bases · The Apollo Computer · Space Navigation · A Video Game of Space similar to Mine Craft But I have a question that relates to space. And is more serious than it sounds: When the Enterprise goes warp speed how does it avoid objects in space, such as comets, dust, and planets? Also, the Enterprise shots photon torpedoes without a reaction force sending the ship with an opposite force. I don’t understand photon torpedoes, but wouldn’t it be simpler to send a torpedo at warp speed, that instead of causing a small part of the ship to be damaged, would totally obliterate the enemy ship. This torpedo would act as F = ma and the resulting energy release would destroy everything. Also, the shields can’t stop such a torpedo. There are “shield harmonics” (like in sound) that block the energy of phasors. Stopping a solid object would be more difficult. So, lasers and phasors are blocked, but traditional projectiles such as missiles are not. Finally, when a space ship is hit by a projectile, I would like to see it spin end over end out of control in space, before the propulsion engines could make the necessary adjustments. I know there is a book called “The Physics of Star Trek.” It may have these answers. All that I have read so far is the inertia dampers prevent the force of coming out of warp speed from acting on the passengers. Has anyone here read the entire book? I don’t know if my questions are answered there. I shouldn’t question the physics of the ship, since Scotty and Spock make it work. It is shows like these why we are interested in science in the first place.
  11. Prime Products just one last time

    PNP = 85 (((((x^2*PNP^4 + 2*PNP^2*x^5) + x^8)/PNP^4) - ((1 - x^2/(2*PNP))))*((PNP^2/ x^2))) == PNP^2 85 (7225 (-1 + x^2/170 + (52200625 x^2 + 14450 x^5 + x^8)/52200625))/x^2 == 7225 Solve[(7225 (-1 + x^2/170 + (52200625 x^2 + 14450 x^5 + x^8)/52200625))/x^2 == 7225, {x}] {{x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 1]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 2]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 3]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 4]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 5]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 6]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 7]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 8]}} N[%3] {{x -> -24.3484}, {x -> 4.97889}, {x -> -4.2072 - 2.87925 I}, {x -> -4.2072 + 2.87925 I}, {x -> 1.71775 - 5.00069 I}, {x -> 1.71775 + 5.00069 I}, {x -> 12.1742 - 21.0804 I}, {x -> 12.1742 + 21.0804 I}} f[x_] := (((((x^2*PNP^4 + 2*PNP^2*x^5) + x^8)/ PNP^4) - ((1 - x^2/(2*PNP))))*((PNP^2/x^2))) == PNP^2 b = [Integrate[f, {x, PNP - 1, PNP}]] b = f[85] The above equation is simplifying the PNP polynomial equation from SFN post 1. This equation proves to be true but there is no way to solve it. The equation shows where x is as x approaches N (or PNP). Taken this feature and using the integral to simplify the original equation where PNP is 85 (in this example), we know that the integral from N - 1 (that is 85 - 1) to N (which is 85) is equal to f(85). Of course some figuring is wrong here. I haven't taken a calculus class in almost 20 years. I don't know if it is possible to have the integral equal the original function. But intuitively it seems to work. I know fault will be found here. I'm just bouncing off ideas. I would have formatted this post better, but Big Bang Theory is coming on.
  12. Prime Products just one last time

    PNP = 85 (((((x^2*PNP^4 + 2*PNP^2*x^5) + x^8)/PNP^4) - ((1 - x^2/(2*PNP))))*((PNP^2/ x^2))) == PNP^2 85 (7225 (-1 + x^2/170 + (52200625 x^2 + 14450 x^5 + x^8)/52200625))/x^2 == 7225 Solve[(7225 (-1 + x^2/170 + (52200625 x^2 + 14450 x^5 + x^8)/52200625))/x^2 == 7225, {x}] {{x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 1]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 2]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 3]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 4]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 5]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 6]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 7]}, {x -> Root[-104401250 + 614125 #1^2 + 28900 #1^5 + 2 #1^8 &, 8]}} N[%3] {{x -> -24.3484}, {x -> 4.97889}, {x -> -4.2072 - 2.87925 I}, {x -> -4.2072 + 2.87925 I}, {x -> 1.71775 - 5.00069 I}, {x -> 1.71775 + 5.00069 I}, {x -> 12.1742 - 21.0804 I}, {x -> 12.1742 + 21.0804 I}} f[x_] := (((((x^2*PNP^4 + 2*PNP^2*x^5) + x^8)/ PNP^4) - ((1 - x^2/(2*PNP))))*((PNP^2/x^2))) == PNP^2 f'[x] (7225 (x/85 + (104401250 x + 72250 x^4 + 8 x^7)/52200625))/x^2 - ( 14450 (-1 + x^2/170 + (52200625 x^2 + 14450 x^5 + x^8)/52200625))/x^3 == 0 g[x_] := f'[x] 1 + (85 - f[1])/f'[1] False 313875685/208889212 N[313875685/208889212] 1.50259 Again the above equation has 5 as a solution knowing only N. I know this is not enough until tried with all values, but it seems that Mathematica has a solution for this instance. I want to show what I was trying to do at the bottom (end) of the equations. I was trying to use Newton's Method. I studied the equations from Wikipedia and felt that maybe applying the method would simplify my equations. I used an y of 85 (a y of N, instead of zero) and placed the slope (the derivative of my PNP equation) in order to solve a test value of x. This test value x is intended to be an estimate, however I tried to find a modified the equations to find x based on a given number (start point; 1) and the slope (derivative) of the original equation (from the first post of this thread). I know this didn't work, but I share just to walk through the idea. If there was a way to find a given value of an equation using the equation and its slope, it would solve my original equation. Below is the equation. 0 = f ' (xn) *(x - xn) + f(xn) 85 = f ' (xn) *(x - xn) + f(xn) xn = 1 ---- This is the start value. 85 = f ' (xn) (x - xn) + f (xn) 1 + [85 - f(xn) / f '(xn)] = x I am aware this doesn't work. However I thought the idea was so simplistic it could work. This is what I am proposing: The slope is known and any start number can be used, so finding the value of x where the y-coordinate equals N can be found without the complexity of the original equation. Obviously, it has been done before. But I am asking for help in trying to apply it to my equation. 20170916SFNnewtonMethod008b.nb
  13. Prime Products just one last time

    Yes, imatfaal is correct those patterns are true for all numbers. I intend to redeem the mistake. Early in my work I was just finding different ways to represent “y” in terms of “x”. It led to a lot of x = x. But I did have several equations that were distinct and significant. Below are some patterns that are redundant, but I encourage you to look because if I do find one that is distinct it is gold dust. I believe the equation at the start of this post to be distinct. I know it cannot be solved. But I think it is a pattern. Yes, my equation solved N knowing x, which wasn’t significant. But if there a way to solve the equation N would be the only number needed. So here we go again: PNP= 85 x=5 {(((((PNP^4)/x)+2*(PNP^2*x^2)+x^5)/PNP^3)* x^3) / PNP } 85 5 {2160900/83521} N[{2160900/83521}] {25.872535051065`} Sqrt[25.8725] 5.0865 PNP= 85 x=5 {((PNP^4+2*(PNP^2*x^2)+x^5)/PNP^3)} 85 5 {420520/4913} N[{420520/4913}] {85.5933} N[{2102600/4913}] {427.967} 427.966619173621`/85 5.0349 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------___ ___ ___ ___ ___ ___ ___ _ Other Patterns related to above work 125/5 = 25 25* 52200625 = 1305015625 --------------------------- 52200625/614125 =N ___________________________________ 52200625 / 125 = 420520 _______ 85^3 = 614125 614125 / 125 = 17^3 = 4913 ________ 83521/ 4913 = 17^4 / 17^3 = 17 _____________________ 2102600 / 420520 = 5 MaybeSIGSFN20170813V02.nb
  14. Prime Products just one last time

    You are correct. That paragraph is garbage. I am not sure what I meant at the time I wrote it. These calculations will always be a perfect square root. Thus finding the product of 2 Prime numbers is equivalent to this equation being used to find a perfect square. Yes other products will also have a perfect square, but it will be a decimal and not a whole number These calculations will always be a perfect square root. I mean the patterns I just showed are always a square root. Which there is no arguments about. It may seem this is of no use, but I am only showing a pattern; A series of patterns that give use a feel for what is happening in the factorization. Thus finding the product of 2 Prime numbers is equivalent to this equation being used to find a perfect square. In the succeeding patterns of the first write-up only 85/7 equals a whole number. I know this again does not seem to prove anything, but here I am only listing patterns. I will give a better explain my example. Testing to see if 3 and 28.3333 are Prime multiples of 85: 1/(85 * 3) = 1/(0.0039215) = 255 28.3333^2 = 802.778 255 - 802.778 = -547.778 -547.778/ 28.3333 = -19.3334 28.3333 / 19.3334= -1.46551 1/-1.46551 = -0.682355 -0.682355 * -547.778 = 373.779 Sqrt[373.779] = 19.3334 28.3333 / -547.778 = -0.51724 19.3334/-0.51724 = -373.791 Sqrt[373.791] =19.3337 This is true that 19.3337 is not a whole number. Obviously. I know that 5 * 17 = 85 and I will square 17 in part of the following example. Obviously if I know this it doesn’t help me solving knowing only 85. I am only trying to find out what is happening in the factorization. But remember that: 1/(85 * 3) = 1/(0.0039215) = 255 255/(85 / 3) = 255/(28.333333) = 9 Sqrt[9] = 3 But it is interesting that 9 + 19.3337 approximately = 28.3333 It must be tested for all numbers and different potential Prime factors but know: 17^2 = 289 425 - 289 = 136 136 / 17 = 8 17 / 8 = 2.125 1/2.125 = 0.470588 0.470588 * 136 = 64 Sqrt[64] = 8 8/0.125 = 64 Is unique among other factors. It is the correct factor. This pattern does not solve that, but shows something interesting is going on. We must test for more for more numbers, but the same is true for 85/11. I am not solving a pattern in Prime numbers. I only care what happens when factored. I know it doesn’t seem like much but it is how I visual the equations. For example the factors of 85 have to be less than 85. And when you choose the smaller factor the second largest factor is limited in value. I know it has been tried and there are already algorithms. But I just want to show that my idea is unique. And I realize it uses test values. But these patterns are important in understanding how I derived my equation. The one that is ugly and slower than division. But again, what patterns are going on in the calculation of that equation. That is why I shared this. To reiterate: 85 *11 =935 935/(85/11) = 121 Sqrt(121) = 11 85/11 = 7.72727 Abs[(7.72727^2 - 935] = 875.293471 875.293471/7.722727 = 113.27322731 Abs[113.27322731 -121] = 7.7267 The solution of 7.7227 is similar to what happen with 85/3 as 28.333. The square 9 for 3 and square 121 for 11 shows a pattern in the factoring. That is if I just didn’t redundantly cause the pattern. I hope it is understood what I tried to present.
  15. Prime Products just one last time

    You should read further. I am aware of the simple pattern that cancels each other out. I canceled many of equations looking for the right one. 1/(85*3) 1/0.0039215686 = 255 255/(85/3) = 9 I know these patterns don’t seem like much, but they were the start of my last equation. The one that you claim will give factors but is slower than division. I argue it gives a position (a distance) from the Prime product. But I am not working on that here I just want to show patterns. I know this isn’t the best written explanation. And I know I have trouble describing my ideas to others. But if you decide to read further down the line there should be a pattern. It has been 6 years since I wrote this. I have to study it myself, because as I stated in my previous post, I have lots of data. Start at the 3rd paragraph of equations. Where 17^2 = 289 for relevant content. Remember = sign means they are equal, but the line below is a pattern where I mixed the numbers together. The below line is not always equal to the top. It is its own operation. This is probably bad practice. But I do not know of a better way to write it down.