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You are not logged in. #1 20120607 21:05:04
Polygonal Number Counting FunctionLet polygonal numbers of order greater than 2 be defined as the various different numbers: , when integers and are greater than , and let represent how many such numbers there are less than or equal to a given number . Then, where: and where: is the "Blazys constant", which generates all of the prime numbers in sequence by the following rule: Integer part of is Integer part of is Integer part of is (and so on...) The following table represents approximated by . ______________________________________ _____________Difference 10_______________________3______________________5___________________2 100______________________57_____________________60__________________3 1,000____________________622____________________628_________________6 10,000___________________6,357__________________6,364________________7 100,000__________________63,889_________________63,910_______________21 1,000,000________________639,946________________639,963______________17 10,000,000_______________6,402,325______________6,402,362_____________37 100,000,000______________64,032,121_____________64,032,273____________152 1,000,000,000____________640,349,979____________640,350,090____________111 10,000,000,000___________6,403,587,409__________6,403,587,408__________1 100,000,000,000__________64,036,148,166_________64,036,147,620_________546 1,000,000,000,000________640,362,343,980________640,362,340,975________3005 10,000,000,000,000_______6,403,626,146,905______6,403,626,142,352_______4554 100,000,000,000,000______64,036,270,046,655_____64,036,270,047,131_______476 200,000,000,000,000______128,072,542,422,652____128,072,542,422,781______129 300,000,000,000,000______192,108,815,175,881____192,108,815,178,717______2836 400,000,000,000,000______256,145,088,132,145____256,145,088,130,891_____1254 500,000,000,000,000______320,181,361,209,667____320,181,361,208,163_____1504 600,000,000,000,000______384,217,634,373,721____384,217,634,374,108______387 700,000,000,000,000______448,253,907,613,837____448,253,907,607,119_____6718 800,000,000,000,000______512,290,180,895,369____512,290,180,893,137_____2232 900,000,000,000,000______576,326,454,221,727____576,326,454,222,404______677 1,000,000,000,000,000____640,362,727,589,917____640,362,727,587,828_____2089 Now, if we use the last 10 values of and to solve for , and then inject those values of into the expression: as goes to the results will be as follows: _____________________________________________________________________ 100,000,000,000,000______64,036,270,046,655_____2.5665438294154____137.03599916477 200,000,000,000,000______128,072,542,422,652____2.5665438318173____137.03599909419 300,000,000,000,000______192,108,815,175,881____2.5665438266710____137.03599924542 400,000,000,000,000______256,145,088,132,145____2.5665438340142____137.03599902963 500,000,000,000,000______320,181,361,209,667____2.5665438339138____137.03599903258 600,000,000,000,000______384,217,634,373,721____2.5665438318063____137.03599909451 700,000,000,000,000______448,253,907,613,837____2.5665438377183____137.03599892078 800,000,000,000,000______512,290,180,895,369____2.5665438337865____137.03599903632 900,000,000,000,000______576,326,454,221,727____2.5665438317301____137.03599909675 1,000,000,000,000,000____640,362,727,589,917____2.5665438334003____137.03599904767 Taking the average of the column results in: , which is an exellent approximation considering that we used only 10 samples from relatively low values of , and taking the average of the column results in: , which is very close to the most precisely determined value of the fine structure constant to date, and matches the latest Codata value perfectly! So, in theory, if we had sufficiently large values of , say , to about or so... then we can simply take the average of sufficiently many random samples of to get to as many decimal places as we like, and thereby generate the entire sequence of primes in sequential order! It's essentially the same principle as flipping a coin sufficiently many times and averaging out the results in order to get as close to as we like. I really like the idea of using one erratic sequence to generate another. It's kind of like fighting fire with fire. Don. #2 20120607 22:20:34
Re: Polygonal Number Counting FunctionHi Don Blazys; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #3 20120610 21:01:25
Re: Polygonal Number Counting FunctionThanks bobbym, The above counting function can be used to approximate the answer, but the exact value of remains a mystery. Why does approximating the number of polygonal numbers of order greater than 2 to a high degree of accuracy require the "running" of the fine structure constant which is by far the most important constant in all of physics? Again, nobody knows! Google searching the phrase " reflexive polygons in string theory" brings up all kinds of results showing that polygonal numbers are at the very core of string theory, but so far, that entire issue remains a mystery! __________________________________________Math is challenging._____________________________________________ Everybody loves a challenge. Indeed, people have climbed Mt. Everest and swam across the English Channel simply because it was a challenge and for no reason other than "it was there". A life without challenges is dull, boring and hardly worth living while a life that is filled with challenges is extraordinarily interesting and (most importantly), loads of fun! The counting function in post #1 is a perfect example of just how challenging some math problems can be. Seperating the polygonal numbers of order greater than 2 from the rest of the polygonal numbers is analogous to seperating the composite numbers from the prime numbers. Both are extraordinarily hard to do, and doing either results in sequences that are absolutely random and erratic, yet follow certain other laws in a manner that is quite predictable. Polygonal numbers of order greater than 2 have only been counted up to . That's the current "world record". A lot of coders tried very hard to break that record, but most of them gave up after their computers either crashed or ground to a halt. However, I'm sure that other coders will continue trying to break that record, not only because breaking records is a fun and challenging thing to do, but because the counting function in post #1 is perhaps the most unique counting function in all of mathematics, and as such, gets first page ranking by Google and is even referenced in the Online Encyclopedia of Integer Sequences. It is certainly the only counting function that involves polygonal numbers. I put it here, just in case you might want to try and break that record. If you don't, then please lock this thread and I will continue having fun elsewhere. Cheers, Don Last edited by Don Blazys (20120610 21:04:25) #4 20120610 21:26:42
Re: Polygonal Number Counting FunctionHi Don Blazys; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20120610 23:09:53
Re: Polygonal Number Counting Functionhi Don,
My problem is with that innocent "and so on" but that gave some negative values so I adjusted to Better. I got 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and I'm thinking 'hey, this is interesting!' but then it all went haywire with 40, 84, 347, 431, 479 ......... Whoops. So please would you make your generator clearer for me. Thanks, Bob You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #6 20120610 23:37:16
Re: Polygonal Number Counting FunctionHi Bob; which is good up until the last entry of 148. That Blazy constant is only an approximation up there and must eventually fail. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20120611 00:47:13
Re: Polygonal Number Counting FunctionI see. I'm not sure Excel will work that accurately, but I'll give it a try. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #8 20120613 17:30:43
Re: Polygonal Number Counting FunctionIf we know sufficiently many prime numbers in advance, then it is exeedingly easy to calculate and solving for in the counting function in post #1. This astonishing relationship between prime numbers and polygonal numbers of order greater than 2 really should undergo further testing using even higher values of , and that will require an expert coder with access to a supercomputer. Do you have any suggestions as to where I can find such a coder? Don. #9 20120613 17:44:02
Re: Polygonal Number Counting Functionhi Don, You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #10 20120716 18:39:17
Re: Polygonal Number Counting FunctionQuoting bob bundy:
Thanks Bob. and the only way to determine which of those variations will remain highly accurate is to determine higher values of . Quoting bob bundy:
Google searching "prime number generating formulas" shows that there are many such formulas, Since this is the only known method which generates all the primes and only the primes in sequential order, I think that testing its efficiency would be interesting, informative, and a lot of fun. Don. #11 20120716 20:59:25
Re: Polygonal Number Counting Functionhi Don,
Trouble with this seems to me to be that this search will never end. What I was hoping for is a proof, and I don't see why that needs any computing power at all. Now you could substitute values of n and show it always works for those values. In time you could try ever larger values and it would still work. But I can show it works using algebra and that covers all values of n in one go. Now I still have a question unanswered from post #5. I'll repeat it here: Is your generating formula And if not, then what? Thanks, Bob You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #12 20120719 22:30:33
Re: Polygonal Number Counting FunctionHi Bob,
The primes are generated from as follows:Start with: then: then: and so on. Thus, is correct. Note that the "previous prime" is simply the floor function of the "previous value". Quoting bob bundy:
Carl Gauss discovered the "simple" prime number counting function: while still in his teens. Proving that it works all the way into required another century of hard labor by some of the worlds greatest mathematicians. Now, my counting function for polygonal numbers of order greater than 2 is very, very sophisticated in that it involves not only the above prime number counting function, but and as well! Thus, proving its convergence with would be extraordinarily difficult, and proving that it generates all of the primes and only the primes in sequential order would actually require a proof of the Riemann hypothesis, which may or may not be provable! Quoting bob bundy:
From my point of view, .Thus, if the search will never end, then the fun will never end, and that's a good thing! Don #13 20120720 01:56:33
Re: Polygonal Number Counting FunctionThanks Don, You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #14 20120805 20:16:25
Re: Polygonal Number Counting FunctionThread closed at the request of the original poster. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. 