I see that you guys have been talking about my formula. I would like to clarify a few things.
My formula is not Erastothenes sieve, though it does use a similar approach to the solution. Unlike other formulas for pi(x), which are either a) approximates, or b) accurate only until a certain point, my formula is 100% accurate at solving pi(x) for ALL x values from 1 to infinite!!! My formula uses minimally all primes smaller or equal to the sqrt(x) to give an accurate solution to pi(x). I am not sure if there is an accurate formula for pi(x) which ALWAYS works, if there is than even the math profs at my school have not heard of it. My formula finds out how many primes are smaller than x, but subtracting the number of integers which are not prime smaller or equal to x. So I have found a pattern, which I have written down as a formula, which in effect eliminates all numbers which are not prime in Erastothenes sieve, once and ONLY once. Then I subtract x from that number to give me my accurate solution of pi(x). Unlike Erastothenes, I only have to initially store the primes in memory, not every number smaller or equal to x to find the solution. The solution is so simple, a computer science undergraduate has figured it out, and apparently I am the first... unless you guys can tell me of one... though, that would mean the professors at my schools math department are not aware, or at least the prof I am with. I have written a document explaining the logic behind my formula, and what each part of it does, with an example. I have send it to the prof I am working with, and he is setting up a meeting with the number theory mathematician at my university to help me... They still want me to come up with a mathematical proof for my formula, so I do not know how long it will be until it is published, or I can share it with you, without someone trying to steal my intellectual property. The logic of how the formula works is pretty convincing, and you can easily see that it should give an accurate solution to pi(x) for all positive values of x. Another interesting thing about my formula, is that it is "backwards compatible", meaning if you "expand" my formula to solve pi(500), then that static formula will also compute pi(x) accurately at least until pi(500). Therefore if infinite was theoretically a concrete number, and you expanded my formula to calculate pi(infinity), then you could use that formula, statically, to accurately allow you to calculate all positive calues of x. Though infinite is not a concrete number, so a dynamic formula is the best I have come up with. Maybe once my discovery is out, someone else will find a better solution. I have cut down the time it takes for me to calculate pi(x) by a factor of 2 this week.... simplifying my formula slightly... though my formula is still an O(n) solution, so it is slow in the long run. Though it calculates pi(100,000,000) in 3 seconds, and pi(1,000,000,000) in 30 seconds, when counting the primes starting the count at 2 up to even 5,000,000 took my notebook 20 seconds. My formula is at least an improvment on that method. Though I do believe that my formula can be simplified further, and I may end up simplifying it further if I can eliminate some useless terms. I am hoping to simplify it to an O(lg n) solution, but I may not find it. I hope this clears up a few things about my discovery, and I hope to be able to share it with you guys soon, when I publish it, you guys will be the first to know! If you know of any formula's which already does what I am saying please let me know, a lot of the information you have posted has helped me discover more things about primes than I knew before, thank you. I know many of you guys are skeptical, and I hope that when I publish this formula you will see that I am for real, and not just making this up. By the way my formula comes up with the following solutions of pi(x) for the following: pi(1,000,000,000) = 50,847,534 pi(100,000,000) = 5,761,455 meaning there are 45,086,079 nine digit primes. Either way hope to hear more about this subject, from you guys, and comments, and any help or resources or places to look for ideas would be nice. I will try and keep you guys up to date on my progress, thanks for responding. Allen Poapst, 3rd year Computer Science, Brock University _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
