- **Octoproth Search**
(*https://www.mersenneforum.org/forumdisplay.php?f=63*)

- - **Number of octoproths per n**
(*https://www.mersenneforum.org/showthread.php?t=5344*)

Number of octoproths per nAll ranges checked from k=1 to 2^n (where 2^n-k would be 0).
All primes verified with PARI 2.1.7. n<=26 op=0 n=27 op=1 n=28 op=2 n=29 op=1 n=30 op=1 n=31 op=2 n=32 op=6 n=33 op=2 n=34 op=13 n=35 op=26 n=36 op=11 n=37 op=92 n=38 op=28 n=39 op=83 n=40 op=331 n=41 op=110 n=42 op=453 n=43 op=632 n=44 op=1297 n=45 op=2129 n=46 op=5017 n=47 op=5278 n=48 op=3979 n=49 op=56905 n=50 op=18547 n=51 op=16870 n=52 op=219117 n=53 op=60620 n=54 op=230143 n=55 op=786971 n=56 op=285415 Full list of Octoproths in completed ranges (up to n=54): Raw text: [url]http://octoproth.greenbank.org/downloads/octo_complete.txt[/url] 12024KB (12MB!) ZIP: [url]http://octoproth.greenbank.org/downloads/octo_complete.zip[/url] 4922KB (4.9MB) TODO - Download link for user discovered Octoproths (with attributions) |

Very good work!
It is very interesting that we can predict the total number of octoproths for a given n!!! I've worked out it today: by modifying some very hard conjectures, first I define for every n value the "weight" of n: in (PARI): [CODE] w(n)=T=128.0;forprime(p=3,10^4,l=listcreate(8);g=Mod(2,p)^n;h=1/g;a=[g,-g,h,-h,2*g,-2*g,h/2,-h/2];\ a=lift(a);for(i=1,8,listput(l,a[i],i));l=listsort(l,1);T*=(1-length(l)/p)/(1-1/p)^8);return(T)[/CODE] Then using it we can predict the total number of octoproths for a given n value by: [CODE] f(n)=floor(w(n)*2^n/(n*log(2))^8*1/16) [/CODE] Try it! For n=51 it gives that f(n)=16537 It is a very good approximation because Greenbank has calculated that the true number is 16870 ps you'll need also w() to use f() Note that in w() the w(n) is also a prediction because it is using primes up to 10^4 ( to become faster the computation) |

Nice work.
f(49) = 55410 real count is 56905. |

[QUOTE=R. Gerbicz]Very good work!
It is very interesting that we can predict the total number of octoproths for a given n!!! I've worked out it today: by modifying some very hard conjectures, first I define for every n value the "weight" of n: in (PARI): [CODE] w(n)=T=128.0;forprime(p=3,10^4,l=listcreate(8);g=Mod(2,p)^n;h=1/g;a=[g,-g,h,-h,2*g,-2*g,h/2,-h/2];\ a=lift(a);for(i=1,8,listput(l,a[i],i));l=listsort(l,1);T*=(1-length(l)/p)/(1-1/p)^8);return(T)[/CODE] Then using it we can predict the total number of octoproths for a given n value by: [CODE] f(n)=floor(w(n)*2^n/(n*log(2))^8*1/16) [/CODE] Try it! For n=51 it gives that f(n)=16537 It is a very good approximation because Greenbank has calculated that the true number is 16870 ps you'll need also w() to use f() Note that in w() the w(n) is also a prediction because it is using primes up to 10^4 ( to become faster the computation)[/QUOTE] This may be stupid, but how do I enter the script on PARI (not the hard way, please) |

[QUOTE=fetofs]This may be stupid, but how do I enter the script on PARI (not the hard way, please)[/QUOTE]
I know only one way at the end of line type \ |

55Greenbank
Will take on n=55 Regards Robert Smith |

[QUOTE=R. Gerbicz]I know only one way at the end of line type \[/QUOTE]
Typing IS the hard way! |

55 almost thereGreenbank
I have done 55 up to 3e15, so a little more to do overnight. The file is enormous, how do I get it to you? Maybe you can send me a private message with your email address and I will send. Regards Robert Smith |

2^55 = 3.6E16
So if you've done to 3e15 then you've only done 8.3% |

[QUOTE=Greenbank]2^55 = 3.6E16
So if you've done to 3e15 then you've only done 8.3%[/QUOTE] No! 2^55>3.6E16 You have to use a larger number than -1+2^n, if you want to search the full range for n. I think Robert444444uk correctly search this interval for n=55, but he mistyped here ( just seeing previous submissions from him ), 3e15 isn't a large range for him. And you can also use my formula to give a prediction for the total number. Using my formula it'll be about: f(55)=772430 ( if we are using rounding instead of floor ) So the size of the file will be larger than the size of all previous files altogether ( up to n=54 ) |

Yeah, I meant to say 2^55 ~ 3.6E16. I was just trying to point out that 3E15 was an order of magnitude out.
2^55 = 36028797018963968 3.6E16 = 36000000000000000 robert44444uk, I'll send you an email about getting the n=55 stuff. |

SorryNo, you guys are totally right, and I was totally wrong. I did not notice until this morning as I set my computer on its merry way for what I thought was the last 0.6E15, and realised I was a whole magitude adrift.
But I doubt I have the computer resources to run this much further. Each 1E15 takes about 8 hours, so the whole exercise will take someone 10 days. I will complete to 4E15, and then release this one. Regards Robert Smith |

[QUOTE=robert44444uk]No, you guys are totally right, and I was totally wrong. I did not notice until this morning as I set my computer on its merry way for what I thought was the last 0.6E15, and realised I was a whole magitude adrift.
But I doubt I have the computer resources to run this much further. Each 1E15 takes about 8 hours, so the whole exercise will take someone 10 days. I will complete to 4E15, and then release this one. Regards Robert Smith[/QUOTE] If I put my whole G5 on it I can have n=55 done in a shade under 2 days. I think this may be the limit of computing all Octoproths for a specific n. Let me check Robert's estimation formula:- est(54) = 225950 (actual = 230143) est(55) = 772429 est(56) = 280214 est(57) = 1253015 est(58) = 2335233 est(59) = 2922982 est(60) = 5869985 I might do n=56 as well, since it has fewer Octoproths than n=55. n=57 looks like a step too far. |

[QUOTE=Greenbank]
est(54) = 225950 (actual = 230143) est(55) = 772429 est(56) = 280214 est(57) = 1253015 I might do n=56 as well, since it has fewer Octoproths than n=55. n=57 looks like a step too far.[/QUOTE] n=55 [0,1E16] = 229947 n=55 [1E16,2E16] = 214250 n=55 [2E16,3E16] = 211102 n=55 [3E16,3.61E16] = 131672 Every 3-prp output by your program was indeed a prime! None failed the tests with PARI. 229947 + 214250 + 211102 + 131672 = 786971 est(54) = 225950 (actual = 230143) est(55) = 772429 (actual = 786971) |

OK, n=56 is complete!
Estimate was 280214. Final count after isprime() PARI checking script: 285415 2^n = 72057594037927936 so I used a kmax of 72060T. Output: n=56, kmin=0T, kmax=72060T, version=5.0, here T=10^12 Starting the sieve... Using the first 11 primes to reduce the size of the sieve array [285415 lines removed] The sieving is complete. Number of Prp tests=1503700301 Time=187990 sec. So that's 2 days, 4 hours, 13 minutes, 10 seconds. That's the end of it, not going to start n=57, there are too many of them! |

[QUOTE=Greenbank]OK, n=56 is complete!
Estimate was 280214. Final count after isprime() PARI checking script: 285415 [/QUOTE] That's great! [Code] g(n)=round(f(n)*(1+1/n)) [/CODE] I don't know why but it seems a better formula for large n values! Try it. |

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