20110219, 19:07  #1 
Feb 2011
163 Posts 
Standard crank division by zero thread
Here:
httр://donblazys.com/on_рolygonal_numbers_3.рdf you will find my "Special Polygonal Number Counting Function" that approximates (to a very high degree of accuracy) how many "polygonal numbers of order greater than " there are under some given number in much the same way that the function: approximates how many primes there are under some given number . The reason that I am posting it here is because this forum seems to have many experienced "coders" who have access to some very powerful computers. Here is my question... Would it be possible to calculate to say, or so? Given that (the number of primes under ) has been calculated to , I should think that this would be "easy", or at least, "possible in a reasonable amount of time", but as it turns out, the coders who determined the present "world record" informed me that determining would probably take about a year. There is an intrepid young coder who is letting his computer run constantly and will verify that "world record" in a few days from now. He estimates that he will be able to determine in about 6 or 7 months, so I doubt that his machine is powerful enough to determine within our lifetimes! I don't own a computer (I use my grand daughters laptop to post) nor do I know anything about "coding" or "programing". Thus, I would greatly appreciate any help or advice whatsoever as to how a determination of might be possible. Thanks in advance, Don. 
20110219, 23:18  #2  
"Forget I exist"
Jul 2009
Dartmouth NS
2·5^{2}·13^{2} Posts 
Quote:
I know my values may be off so first I''l check them up (no I'm not a mathematician), mass of proton according to my calculator (which if I'm not mistaken is needed for the mass ratio you talk of) is about 1.67 *10^27 , mass of electron is about 9.109 * 10^31, the ratio my calculator gives me with the ratio of the 2 constants is 1836.15 but I'll shorten that to 1.83*10^3 assuming the e talked of that's not defined is 2.71828............ ? , and you give the fine structure constant. I find that is about 9.99**10^1 which if my math is correct means the figures in the table are wrong. never mind I don't see why the first equation then but like I said I'm not mathematical Last fiddled with by science_man_88 on 20110220 at 00:01 

20110221, 00:44  #3 
Feb 2011
163_{10} Posts 
The counting function works.
In fact, it approximates the number of polygonal numbers of order greater than 2 to a much higher degree of accuracy than approximates the prime numbers. The challenge now is to determine to about . That will tell us a lot. Don. 
20110221, 01:30  #4  
Aug 2006
2^{2}·3·499 Posts 
Quote:
Your approximation is constant * n + lower order terms and your paper shows you modifying the coefficient of n based on counts done so far. That you use constants from physics is of no particular significance here; higher counts will show that the constant term is still off and needs to be modified. Now I'm curious about this "young coder" you have working on the task. What algorithm is he using? Based on the speed it seems like direct enumeration, which would seem to be a very slow method for the task. I would think that the theory of Diophantine equations could be used to construct an inclusionexclusion method that would be orders of magnitude faster. Last fiddled with by CRGreathouse on 20110221 at 01:31 

20110221, 15:09  #5  
Feb 2011
163 Posts 
Quoting CRGreathouse:
Quote:
which can be found here: http://oeis.org/A090466 is not a "well behaved" sequence. As with the sequence of primes, it can only be described as erratic, irregular, patternless, random and unpredictable. The random fluctuations in may not be as pronounced as the random fluctuations in (the number of primes under x) but they are nevertheless quite random, and the sequence is extraordinarily difficult to count. At least a dozen coders crashed their computers trying to break the present "world record" . Quoting CRGreathouse Quote:
, and since I am the first mathematician to develop a "counting function" for "polygonal numbers of order greater than 2", I suppose that I could name the coefficients .6403627... and .4001125... after myself and call them "Blazys constants", but I am much too humble and modest to do that! Quoting CRGreathouse: Quote:
there was very little data to work with. However, as the coders provided me with higher counts of . the "physical" constants and emerged naturally. That's why I am quite certain that the function is correct, and will hold up regardless of how high the counts are. Quoting CRGreathouse: Quote:
Since I don't know anything about computers or coding, I really can't comment on his methods, but he recently informed me that will be determined by this coming Friday, and that will be the new world record. If you are interested, I will post the result here as it comes in. Don. 

20110221, 15:26  #6  
"Bob Silverman"
Nov 2003
North of Boston
16606_{8} Posts 
Quote:
He said that the polynomials were well behaved. Quote:
Quote:
Quote:
have computed values. Please show us the derivation of your counting function. Or is merely an emprical result from fitting curves? 

20110221, 20:07  #7  
Aug 2006
2^{2}·3·499 Posts 
Quote:
Also, you seem to have only an empirical fit. What have you actually proved? I can prove bounds on the distribution function; can you? Quote:
I'm not prepared at the moment to commit the time or processor power, but at some point I may want to compete with the team of you and your coder to reach 10^14. 

20110221, 21:10  #8  
"Forget I exist"
Jul 2009
Dartmouth NS
10000100000010_{2} Posts 
Quote:
according to my math: Code:
alpha = 137.035999084^1 Code:
micro = 1836.15267247 Code:
blazy(x)=(x(x/(alpha*micro*Pi*exp(1)+exp(1)))(.5*(sqrt(x(x/(alpha*Pi*exp(1)+exp(1)))))))*(1(alpha/(micro(2*exp(1))))) Quote:


20110221, 21:42  #9 
"Forget I exist"
Jul 2009
Dartmouth NS
2102_{16} Posts 
never mind coding error.. added a term in.

20110221, 22:02  #10  
"Forget I exist"
Jul 2009
Dartmouth NS
10000100000010_{2} Posts 
result after correction
the only error I found was a micro where it didn't belong, with this corrected I redid the test (took 0 ms according to my timer) and I got the new result of:
Quote:
Last fiddled with by science_man_88 on 20110221 at 22:22 

20110221, 22:39  #11 
"Forget I exist"
Jul 2009
Dartmouth NS
2·5^{2}·13^{2} Posts 
also with the corrected code and maximum memory allocation my machine can handle in the PARI console I calculated this one:
Code:
(18:24)>blazy(1.2*10^100000000) %34 = 7.684352917150111789071860349 E99999999 (18:30)>## *** last result computed in 41,469 ms. 
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