Trurl

Well I’ve been reading about genetics as a crash course. I am amazed at all the applications that bio-engineering has. But I did notice the intro chapter where there are arguments if bioengineering is safe or ethical. The book tries to be objective, but is really promoting the field. The author figures it has the same pro’s and con’s as any technology. I don’t know that I agree. As for a pattern, I picture a chemistry problem. You don’t mix all the chemicals in the chemistry set together. Chemistry improves our lives dramatically. But that doesn’t mean someone doesn’t take nitrates in an oxygen source and make a bomb. But in the bio-world there would be a one-way function to disarm the bomb. The bomb is easy to make, but not easy to reverse, if the only way to disarm the bomb is to add more chemicals to the mix. In other words, if something is genetically altered and becomes dangerous, how would you reverse it if it multiplies. I am not far enough in my study to find patterns in genes. But I just think of my chemistry example, where it is easy to add to a mixture of chemicals, but not as easy to neutralize the mixture of chemicals, by adding more chemicals to the solution. I have seen interviews on TV of the guy who lead the mapping of the human genome. He claims that he just proved evolution. But I’m not convinced until he can explain all the patterns that exists and cures complex, genetic diseases. So, if the math can be applied, and this isn’t the same math that says it’s a quintillion to 1 chance, like the March Madness bracket, I am going to ask the forum what are the simple strategies of finding patterns in genes? As I learn more I will be able ask better questions.

I think the $100 price tag for 700,000 traits is a deal. I do wonder why 23andme compares to prehistoric man. So, as I understand it, the genetic information is so vast it takes computer processing to find patterns. I don't know what kind of learning curve there would be. However, just by using N = p *q, I found 7+ patterns. All of which we found useless on this message board. But they were patterns even if they didn't solve the prime factoriation problem. Now I am asking where to start looking for genetic patterns. Are there some classifications that are starting points, like N=p*q for RSA? I am good at looking for patterns. I don't have a biology background. But I have read about networked science where amateurs provide an extra perspective working to classify stars or find genetic patterns. So please point me in the direction as to where to have a simple start to look for genetic patterns.

I will start the pseudo code. Notice that an x of 545 is positive and an x test value of 8756 yields a negative. I am arguing that with these test values the difference from PNP shows if the desired x is higher or lower than the test value. There is no reason that calculus won’t give a value when limit (equation) when PNP approaches 0. I know it takes more than that but do you agree the pseudo code will show an indication where x will fall at PNP? Please join this post. If you don’t believe my code, counter it. There should be a pattern in the pseudo-code. x = 545 y = 6737 PNP = 4639* y (((((x^2*PNP^4 + 2*PNP^2*x^5) + x^8)/ PNP^4) - ((1 - x^2/(2*PNP))))*((PNP^2/x^2))) 545 6737 31252943 (566741960869155702888306342808481973/580236226342089968450) 566741960869155702888306342808481973/580236226342089968450 N[566741960869155702888306342808481973/580236226342089968450, 13] 9.767434971132*10^14 Sqrt[9.76743497113228416002242796713336`13.*^14] 3.125289581964*10^7 PNP - 3.125289581963931252118627996719126077423156917`13.\ 301029995663981*^7 47.18036 test second x x = 8756 y = 6737 PNP = 4639* y (((((x^2*PNP^4 + 2*PNP^2*x^5) + x^8)/ PNP^4) - ((1 - x^2/(2*PNP))))*((PNP^2/x^2))) 8756 6737 31252943 73243982077295884748898890760446202375/74884743323939619512464 N[73243982077295884748898890760446202375/74884743323939619512464, 14] 9.7808951231166*10^14 Sqrt[9.780895123116592692523297300191834`14.*^14] 3.1274422653530*10^7 PNP - 3.127442265353046080516318622627538568244967294`14.\ 301029995663981*^7 -21479.653530

Ok so my graphical representation has many flaws. I’m not finished with it yet. But did you guys look at the last attempt? But back to Post # ___16___ and ____19_____ I know it is no longer the Prime factorization problem if you test for values of x knowing N. I argue that with the test value of x you will know if the equation results in the given number N. The closer to the actual x the closer to the given, N. I know you guys don’t like pseudo-code. I am trying to write a program that will test x values. I know we have been down this road before but this is for an education project I am doing. I would explain the details but it would influence your input. I am going to write a program to efficiently test x. If you graph “x” you know where it approaches N. My arguments are in past posts. But I ask to move away from the Prime Factorization to a trail-and-error computer program. Yes, I believe my equation can eliminate calculations on large Semi-Prime numbers. But this is all you need to know for now. Please try and write pseudo-code if not to prove me wrong but to promote education. Thank you for your participation.

https://soundcloud.com/quitteriel/connor-dickiesynbiota-synbio-experiments-for-all What do you think of a kit like this? They start at $45. Also from what you said. Is it safe to share my DNA with Ancestry.com? They give you access to the genome. But would I be able to analyze it?

There are still problems. s does not equal N/Pi but angle AFC should. That is if the triangles are similar. The only place that the sides of the triangle should be an integer is at x and y. AFC = (remainder(N/Pi)) * Pi (In theory)

Ok, sorry I haven’t replied sooner. I have changed that s is now the remainder of N/Pi. That is a 180 the max angle (straight line). I did this because I couldn’t remember the modulus definition. I am saying that if N is much larger than x and y the Prime factors that x would wrap around the half circle (180 degrees) N/Pi times. The remainder would form a triangle where x and y would add to find the remainder of N/Pi as the similar triangle AFC. So triangle AFC is similar to triangle xys. (Yes I know the label isn’t standard.) Why did I do this? There is no guarantee this will make the solution simpler. But I already have proven equations for x and y as related in terms of x and N. I was going to use these equations and place them to the sides of the triangle and use the equations we know about triangles to solve for a pattern. This is a long shot, but I have always pictured the Prime factorization as a logarithmic spiral. That is why I wrapped x at an angle around the circle. I realize this theory has somehow gotten away from me. It is extremely confusing. But I stand behind my underlying theory. As you read my earlier post on this thread, I have found patterns in the Prime factorization. The resulting equations are just impossible to solve to be useful. I thought since I cannot solve the polynomial equation, I would make a graphical representation, but I need to reevaluate this. Perhaps if I map out the entire idea, you guys could assist me in making it makes sense. I am have been busy and it takes time but I think I should explain more, even though this post is an attempt to explain. But in the coming weeks I will work on this. Try not to laugh at me too much.

OK, first off I’m sorry l haven’t replied sooner, but I have been busy. Second, I think you guys overestimate my knowledge of biology and genetics. I want to look for patterns in the genetic code. I have did a little research and I don’t know what the information needs. I need a starting point. You guys gave me excellent information. But I cannot test the DNA myself. I have an upper-level chemistry book that says to place a specimen in a spectrophotometer. I have googled it and it is affordable. My chemistry book does not list steps however. It has theory then puts a list of 7 steps on what to do without a complete lab explanation. I should explain more what I want to do. I want to look for patterns in genetics. As you guys stated that a lot of information is shared. I think it is like a networked science where the public looks at the data, like finding constellations in telescope pictures. I’m not claiming to be able to discover anything. I have tried to find patterns on the Prime factorization problem. I have found some patterns, but I don’t have a solution to the problem. As you can see from my math post on SFN. So I suppose my question is can I do any meaningful experiments. And by meaningful I mean worth the price of a $500 spectrophotometer to do amateur experiments and have some fun. I also have a question of the value of my own DNA. I was thinking of doing the Ancestry.com DNA test. But if I give my DNA do I compromise my privacy? I have DNA on record from being in the military. The military has a great DNA base of every member since approximately 1996. They are only supposed to use the DNA for identification, but imagine the information they collect on everyone with both nature vs. nurture. But I imagine a world where cryptography and identity will utilize the genetic information. Is it safe to give Ancestry.com genetic fingerprint? BTW, when I was talking about God, it was not in a religious way. I was asking does a one-way function exist. If an all-powerful being can create a function can they reverse it? If they can there are no one-way functions. But if the being can’t reverse it, one-way functions exist. Either way some would say that either would mean they are not all knowing. The boulder problem is a non-religious person’s argument. But I mention this only as the meaning of a one-way function. It is not intended to prove any religious viewpoint right or wrong. It is just a conundrum.

https://1drv.ms/i/s!Ao7PhUWlkaBthGPYxf06XrWnNzKL Definitely some problems with my geometric representation. Does this look better?

Ok, for my background I am a college student. I confess to not knowing biology or genetics or even engineering for that matter. I have read some books on genetics and I am particularly interested in gene-therapy. But I have read that genetics is related in many ways to cryptography. I have tried here at SFN in a math post to solve the Prime-product problem. I have had little success so far. I have a polynomial which if you already know the Prime numbers proves true, but by itself I cannot solve the resulting equation. I have tried geometry and am currently looking at other methods. My question is how do you read the genetic code chemically? I have seen some basic experiments in a Make biology book. Also what are the cytological algorithms? From my brief research the math problem is complex and differs from gene pattern to pattern. So not only do you have to solve a genealogic one-way-function, but you don’t know the algorithm that pattern is based on. And most gene knowledge is not shared due to the fact it is a big-money intellectual property. So with most researchers working alone it is no wonder mathematicians haven’t solved it. My approach is to find a pattern that not only describes the RSA Key algorithm, but will find patterns where patterns don’t seem to exist. I believe in creation, but will not preach but list this to note something very important. A nonbeliever asked if God is so powerful can He make a bolder so heavy even he can’t lift. This sounds ignorant. The nonbeliever would say he isn’t all powerful if he can’t lift the bolder and if he can’t lift hit he can’t lift the heaviest boulder. I know it sounds like nonsense. But my question is can God make a function that cannot be reversed? Obviously I don’t know what God is capable of. There are easily one-way functions we cannot reverse. However I don’t see genetics as something that can’t be figured out. Some scientist, see genetics as chemical proof God does not exist. I believe the pattern just shows his work. The fact remains that knowledge of genetics is as dangerous as the fruit from the Tree of Life. That is just my perspective. I don’t want to argue if genetics proves we evolved. I am just stating why I am interested in genetics. I want to find patterns in them. Obviously I am limited in what I can do. I don’t have a lab or biology background. But I found this article which described a one-way biological function. Also you should check out my post in the math forum. Please share if you know any simple genetic experiments. I have the catalog of American Scientific’s “Amateur Scientist” columns. I have a professional microscope, power tools, Internet, cryptography books, and an impossible to solve polynomial. I’m not intending to cure cancer, but I want to look at patterns in DNA that could relate to cryptography. http://www.asee.org/documents/sections/middle-atlantic/fall-2009/01-Biological-One-way-Functions.pdf http://www.scienceforums.net/topic/95813-prime-products-just-one-last-time/

Imatfaal, I am not arguing that 60 deg = 1 radian. 1 rad is 180/Pi. I am only looking to find a symbolic value of angle AFC. The entire point of drawing the triangle with side N is to find angle AFC. As for degrees, it would be fine to find angle AFC in degrees. The question is can anyone symbolically find this angle. If you can I will tell you why I want to know its equation for.

https://1drv.ms/i/s!Ao7PhUWlkaBthFy7UgDHUGEIh2Lm Here is my updated drawing. It could still be wrong. I may have some flawed logic. But I am seeing this idea develop, but I cannot explain it. It is abstract. I know if I write it out and keep it as simple as possible, maybe someone else will be able to see it also. That is what I hope. Let me know what you think. This is only the start of the problem. But I need some input that the vector addition with sides x and y (the 2 Prime numbers multiplied together with an angle between them (the angle approaches the limit of Pi radians) have a resultant of N (where N is the product of the 2 Prime numbers x and y). Let me know if this conveys anything. An idea is not as valuable when no one else can understand what you are trying to do.

Very interesting. The question to answer is how it differs from a regular gear. Inter-locking the forces on the gear could be distributed differently. I am not sure of the inter-locking would allow circular motion. They would be in conflict during rotation. However don't be deterred. If certain Pokadoves were stationary while others rotated as a gear you could produce custom orbits. That is irregular non-circulat orbits. Think unsemetric objects like a space station. This is good work. I'm just giving my opinion. I am no authority on the subject. But you need to research gears in a machine design book. Again cool shape; creative design; I'm just not sure how it moves. Are you suggesting to use it as a mag wheel? Or is it to inter-lock as a tool?

Yes, my drawing and explanation are off. But don’t disregard the idea yet. This is a one of the graphical representations of my algebraic work posted earlier in this post. Things to remember when looking at this drawing: This is a vector. Both x and y are Prime numbers We are looking for the smallest lengths x and y possible. (They represent Prime numbers after all. There are no angle sides multiples of x and y. Again, these are prime factors. So, there is no 2x or 2y or 3x and 3y. As the lengths of x and y increase, the angle between them decreases. This is significant because the vector addition of x and y is less than N. And for my solution x < y. sin(60 degrees) = 0.866025404 sin(1 radian) = 0.841470985 cos(60 degrees) = 0.5 cos(1 radian) = 0.540302306 This relates to the same error of my equations. I am working on this problem off and on. I will try and produce a better and corrected drawing.

Ok, my last post wasn’t very clear. Here is what I was trying to do. N is known. x and y the 2 number that when multiplied together make N. I am trying to simplify the factoring by using a simple vector. This vector might be able to be solved to find the unknown factors x and y. Picture an angle between 2 lines. As the lines increase in distance so the angle stays the same, but the distance between the 2 lines also increases. So I start with a triangle whose segment opposite the (obtuse) angle between the lines equals N. Now I take this same length N and travel along the original 2 lines until the length along the lines approaches a segment between those lines as N. N is the point where the segment between the 2 lines will reach the limit of N. Beyond N the segment is bigger than N. This is where I need your help. Just to see if this is worth pursuing. The other limit (opposite end of the N segment) is 1 Radian with a radius of N. So, on a circle with radius N with 2 sides N and the angle between 1 Radian. I am not sure as you pointed out the triangle of 60 degrees is not 1 radian. But I am referring to the relative angle between side N and side N. So, the tangent of x equals N -x. N is known This is the tricky part! I have not solved this yet. But I am saying I could possibly use the equations I found a pattern in the Prime factors. Which until know is too complex. But Knowing N and the angle opposite N… I am saying there is a possibility of substituting the equations for x with a new simpler equation of x in this geometry. I am not claiming this works. It was just something I was thinking about. Remember this drawing is off. I need to make a clearer drawing. https://1drv.ms/f/s!Ao7PhUWlkaBtgQd7IjIjxkBjv3wz

I am going to keep this short, because it is just an idea I was working on and I have school work to do and not silly math ideas. But imagine a circle where N is the product of 2 Prime numbers. The radius of the circle is N, so in 1 radian a triangle is formed with a segment opposite the 1 radian angle is N. So an equilateral triangle is formed with all sides equaling N. But we want to know the angle of a triangle with the side opposite the obtuse angle is also of length N, but we have lengths "x" and "y" that are unknown sides. Could we take the N-equilateral triangle and subtract x from one of the end sides? The theory is that the y side would complete the triangle giving us the obtuse's angle in form of an equation with variables in N and x. Can anyone disprove that this will not geometrically solve an unknown triangle of one side N, which is known, and put into equation form x and y? I will add more to this when I have time. I can't insert the picture so it is difficult to visualize what I am describing. I know it sounds stupid, but there is some thought here. Let me know what you think. Trurl

Thanks for the link. The following is why I started researching this idea. It may seem silly but does anyone get anything from this? I'm not sure if anyone got anything from the previous posts but this is going to sound brilliant or a bunch of malarkey. Let me start by saying that this idea is just a theory and nothing is proven. This is just an example of how I approach a math problem. It is very much intuitive and visual. The idea: PNP is given and the unit circle will be used to find x and y. (These variables correspond to my previous equations.) The area encompassed by the angles on the unit circle will be used to form equations that find the values of x and y. Imagine a vector solved by the method of parallelogram addition. That is there is a triangle with 2 known sides x and y. Through vector addition and an angle of y/x radians between the sides the area can be found. This area is theoretically equivalent to N in value. The problem is N is the only value we know. To be useful we have to know y/x are at least some properties and proportions. We cannot use vector addition directly because N is only given. But it may be useful to use the unit circle to find which x and y will equal N. So a circle with radius x and a y arc length from an angle of y/x radians, has an area encompassed by the unit circle equal to N. This encompassed area is of angle y/x. If y/x is larger than 2Pi*x the angle encompasses one or more of the circle. That is the idea. I don’t know if it works, but it is how I go about math problems. This would relate to my theory of a logarithmic spiral that would show a pattern in the placement of Prime numbers. I have had this idea for quite some time. As seen in these URL’s: http://www.constructorscorner.net/ideas_and_gadgets/math/math_hunch/hunch_00001/hunches_section0005/trig_parabola.html http://www.constructorscorner.net/ideas_and_gadgets/math/math_hunch/hunch_00001/hunches_section0005/trig_parabola_verified.html http://www.constructorscorner.net/ideas_and_gadgets/math/math_hunch/hunch_00001/hunches_section0008/PrimeRevolutions.html If it is true this is just one step. There must be a way to solve the unknowns x and y in equation form. Remember PNP/y equals a distance of 1 radian. And remember this is only a preliminary idea I want input on. I am not clamming it works. I just think it is interesting enough to consider. Trurl [bJS1]

I don't know the speed. But Mathematica can draw graphs fast. I am looking for some guidance on how to program numbers of hundreds of digits. See for yourself on the code and attachments that these 2 equations work with large values. So if you agree or disagree just post here. I am researching ways to use calculus on the graphs and also have other methods of solving the polynomials. Here is my results of my last effort that has changed from finding patterns in Prime multiplication to solving the found polynomial equations. In[32]:= PNP = 7727* 65537 t = 7727 Sqrt[(((((t^2*PNP^4 + 2*PNP^2 * t^5) + t^8)/ PNP^4 ) - ((1 - t^2/(2*PNP)))) * ((PNP^2/t^2 ))) ] t * ( ((PNP^4/t + 2* (PNP^2 *t^2) + t^5) / PNP^3) - 2*(t^2/PNP) ) Out[32]= 506404399 Out[33]= 7727 Out[34]= (11 Sqrt[18205987286897013994227797/2])/65537 In[36]:= N[(11 Sqrt[18205987286897013994227797/2])/65537, 14] Out[36]= 5.0640530604466*10^8 Out[30]= 142546691485720530013630/281487861809153 In[31]:= N[142546691485720530013630/281487861809153] Out[31]= 5.06404*10^8 SFN_PatternsPDFHigherValues.pdf SFN_PatternsPDFHigherValuesb.pdf

I have attached a PDF of graphs. PlotofUglyEquationsV8_20160704.pdf

Did anyone agree or disagree with my last post? Obviously the odds are against breaking a one way function. But if you feel it doesn’t work please be my guest to call me stupid. Shooting an idea down is better than no response at all. I am serious in my attempts and have other equations or patterns in products. I also need guidance on how to program numbers. I mean recursive factoring. Traditionally large numbers are difficult to program. Mathematica can handle about 300 digits. So proceed to disprove this problem if you decide it needs bashing. Here is another pattern that I believe to be unique of Prime Products. It is simpler than the last, but still complex to solve the polynomial. Many of the equations or patterns as I call them are true for all x and y. However there are some (though complex) patterns are only true to Prime factors as shown below. There are several with one from the last work to this one giving us 2 complicated but true patterns. y = ((PNP^4/x + 2* (PNP^2 * x^2) + x^5) / PNP^3) I believe this equation to be a pattern of multiplication to solve for a y that is unique, that is it doesn't prove always true with all x's. x * y = PNP x * ((PNP^4/x + 2* (PNP^2 * x^2) + x^5) / PNP^3) = PNP or another equation y = y ((PNP^4/x + 2* (PNP^2 * x^2) + x^5) / PNP^3) == PNP/x I will continue to find errors in my equations and post those that are unique to Prime factors. Maybe a simpler mathematical equation can be found. With error : x * ( ((PNP^4/x + 2* (PNP^2 * x^2) + x^5) / PNP^3) - 2*(x^2/PNP) ) = PNP

I do not disagree. I do not have a way to mathematically solve this complex and ugly polynomial. What I am saying is y has been eliminated from the equation. When you graph 85/x to find y both x and y are unknown. The quotient leaves no clue as to what values you are looking for. When you graph the ugly left side of my equation x is still trial and error, but you are looking on the graph for 85^2 (or 85 if you graph the square root of the equation). The graph is a one to one and y increases as x increases. So when an x of 3 is 80 and x of 7 is 88 the x that is 85 is between them. I prose using statics (calculus) to find the properties of the graph. There isn't an advantage for an PNP of 85 but a large factor where factoring is difficult my method may prove useful. But I do not how to program million digit numbers. But do you agree if you use trail and error, you are approaching the known PNP and at least know that your x is larger or smaller than the x which is the factor? That is something division alone does not show.

Wait a minute I decided it was simple logic. In your pen and paper your division does not show how close the calculation was to the correct factor. We know we are aiming for 85. This method shows 3 is below 85 and 7 is above. Do you still think there is no pattern?

I know the equation is ugly. But what if such an equation existed how should it look? I think the one advantage of my method is that it can be graphed. I know that division is faster but I see advantages when working with larger numbers. In your pen and paper estimates you know what to try but my method but mine is static in that it shows how far the tested number is from 85. Yes this has to be proved. This is the reason I share this post. Of course this probably proves wrong. But sometimes you get gold fever and go looking for gold dust.

PNP = 85 x = 3 (((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) 85 3 847772947/130050 N[847772947/130050, 14] 6518.8231218762 Sqrt[6518.82312187620146097654748173779315647828`14.] 80.739229138481 In[1]:= PNP = 85 x = 7 (((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) Out[1]= 85 Out[2]= 7 Out[3]= 5538604027/708050 In[5]:= N[5538604027/708050, 14] Out[5]= 7822.3346190241 In[6]:= Sqrt[7822.33461902408022032342348704187557375892`14.] Out[6]= 88.443963157607 These values do not produce an N of 85. That is, we know the value of N. It is what was eliminated from the right side of the equation. So it is a significant value that is produced by the left hand side of the equation. 3 is 80.7 and 7 is 88.4 so the Prime number with the value closest to 85 (within a fraction of error) is 5. Note the sides of the main equaiton are already squared in form. I did this to remove the square root. The values fo the equation reflect this. A computer program could be written to test values based on the results of the left hand side of the equaiton. I know it isn't a perfect math solutions where I solved the polynomial but there is a distinction in the numbers. It tells the user if the number that was tested was too large or too small. I will address more quesitons in a future post. In[7]:= PNP = 85 x = 5 (((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) Out[7]= 85 Out[8]= 5 Out[9]= 4179323/578 In[10]:= N[4179323/578, 14] In[11]:= 7230.66262975778546712802768166089965397924`14. Sqrt[7230.66262975778546712802768166089965397924`14.] Out[11]= 7230.6626297578 Out[12]= 85.033303062728

The equation is simply: x^2 * y^2 = PNP^2 Which stands for: p^2 * q^2 = N^2 The left side of the equation is x and y. (They are multiplied together to get PNP.) The right side is PNP. Both sides squared of course. I cannot solve this polynomial equation. If I could I would know instantly what x is. However if values are tested starting with smaller Prime numbers, ( x is the smaller product), I believe you will have a feel for where x is because the equation with x plugged in will be approaching PNP (85 in this example). I believe you could statistically use trial and error to find x. (As PNP approaches 85, x approaches 5.) You could change the equations to find y instead of x and use both the y and x versions to statistically eliminate products. The question is does it work. I am not so much concerned with speed. I do not know how to program million digit numbers. But I think the larger the product the more valuable a statistically found solution is. Of course if I could solve the polynomial (Which I can’t now, but I still have a few more tricks in my mathematical tool chest.) the problem would be solved. However I don’t claim this solution to be faster, but it represents a different approach. I do however value feedback. Most people say this is stupid. I agree that the problem is impossible. I just had an idea for a different spin on it. I could use already discovered methods of evaluation, but other than the math lesson that would provide wouldn’t it just be doing the same thing and expecting different results. I approach this as a learning exercise. I have read much about cryptography and Mathematica. In short if it’s wrong, it’s wrong. The only reason I put so much work into it is because I of my first idea that a logarithmic spiral could find a pattern in Prime numbers and maybe other sorts of patterns that seem to have no pattern. So that is what I am trying to do. The only thing that would increase the speed of this equation would be a statistical (calculus) evaluation. If you test for x and it doesn’t equal PNP then you know if it is larger or smaller. Then you test again knowing that when x gets bigger, y gets smaller. So instead of making unlimited processes you can make an educated guess.