- **Conjectures 'R Us**
(*https://www.mersenneforum.org/forumdisplay.php?f=81*)

- - **Bases 2 & 4 reservations/statuses/primes**
(*https://www.mersenneforum.org/showthread.php?t=9830*)

[QUOTE=gd_barnes;122588]OK, I will make note of the test limits and unreserve k=13854, 16734, and 19464. Note that the last one is currently only on the base 16 page because it's a MOB base 4.
Here is what I show for the test limits for the k's that you're unreserving: k=13854 and 16734; I show no work so they are still at n=100K base 4. k=19464 per your status report of 1/7/08 is at n=137112 base 2. Is that correct? I'll coordinate getting the 3 unreserved k's tested. If they aren't reserved in the new few days, I'll reserve and test them. Since you did some testing on k=19464, do you have a sieve file for that one or for any of the k's? If so, I will post them on the website. Gary[/QUOTE] OK for unreserving k=13854 and 16734 ; I am presently sieving for k = 19464, and I wish still test it at least up to 524288 base 2 when sieved enough... Regards, Jean |

[quote=Jean Penné;122607]OK for unreserving k=13854 and 16734 ; I am presently sieving for k = 19464, and I wish still test it at least up to 524288 base 2 when sieved enough...
Regards, Jean[/quote] OK, are k=13854 and 16734 still at n=100K base 4? Gary |

[QUOTE=gd_barnes;122608]OK, are k=13854 and 16734 still at n=100K base 4?
Gary[/QUOTE] k = 13854 is at n = 261944 base 2 (so, 128K base 4). I did'nt worked yet on k = 16734. Jean |

Jean,
I looked at the parity thing you mentioned. It sounds interesting, perhaps we can go to higher bases once we are done with 256. I also looked at all k's 1-32, that are supper fast for all possible parities up to 2^100. There were some really low weight sequences and some high weight sequences. Do you think we might have similar luck like RPS or 15K with some of these high weight or low weight sequences and possibly find a 10M prime? What do you think? If you or any one else is interested please let me know. Thanks:smile: |

Low weight Reisel
K=29 not included as it is already low weight. [code] k*2^m*(2^i)^n-1 // last column is the candidates left, approx estimate of weight 19 9 17 163 19 9 34 332 15 8 35 244 15 28 35 243 19 9 51 496 21 15 55 480 19 2 63 549 21 45 65 416 23 39 65 554 19 43 68 664 21 47 69 613 27 15 69 623 15 1 70 456 15 7 70 384 15 15 70 481 15 21 70 453 15 29 70 483 15 35 70 483 15 43 70 482 15 49 70 483 15 57 70 388 15 63 70 485 15 31 74 457 [/code] |

[QUOTE=Citrix;122746]Jean,
I looked at the parity thing you mentioned. It sounds interesting, perhaps we can go to higher bases once we are done with 256. I also looked at all k's 1-32, that are supper fast for all possible parities up to 2^100. There were some really low weight sequences and some high weight sequences. Do you think we might have similar luck like RPS or 15K with some of these high weight or low weight sequences and possibly find a 10M prime? What do you think? If you or any one else is interested please let me know. Thanks:smile:[/QUOTE] Harsh, Presently, I am more interested in trying to prove one or more of these four mathematical conjectures, than to find large primes (although it may be a subproduct). Moreover, I hope we will not need to find a 10M prime before proving at least one of them! So, perhaps it would be better to restrict this project to base 4 for now, and not to dissipate our efforts too much... But, indeed there is place for other similar projects! Regards, Jean |

[quote=Jean Penné;122480]Hi,
On the 23 May 2006, Citrix warned us, in the Sierpinski base 4 thread, about this problem : [URL]http://www.primepuzzles.net/problems/prob_036.htm[/URL] To be short, the Liskovets assertion is : There are some k values such that k*2^n+1 is composite for all n values of certain fixed parity, and some k values such that k*2^n-1 is composite for all n values of certain fixed parity. It is almost evident that these k values must be searched only amongst the multiples of 3 (the assertion is trivial if 3 does not divide k) : If k == 1 mod 3, then 3 | k*2^n-1 if n is even, and 3 | k*2^n+1 if n is odd. If k == 2 mod 3, then 3 | k*2^n+1 if n is even, and 3 | k*2^n-1 if n is odd. Almost immediately after, Yves Gallot discovered the firt four Liskovets-Gallot numbers ever produced : k*2^n+1=composite for all n=even: k=66741 k*2^n+1=composite for all n=odd: k=95283 k*2^n-1=composite for all n=even: k=39939 k*2^n-1=composite for all n=odd: k=172677 And Yves said that "I conjecture that 66741, 95283, 39939, ... and 172677 are the smallest solutions for the forms - having no algebraic factorization (such as 4*2^n-1 or 9*2^-1) - but I can't prove it." For several reasons, I think it would be interesting for us to coordinate the search in order to prove these four conjectures : 1) They involve only k values that are multiples of 3, so the success will no more be depending of the SoB or Rieselsieve one. 2) For the n even Sierpinski case, only k = 23451 and k = 60849 are remaining, with n up to more than 1,900,000 that is to say there are only two big primes to found, then the conjecture is proven! 3) For the n even Riesel (third line above) there are only four k values remaining : 9519, 14361, 19401 and 20049, although the search is only at the beginning! 4) For the two remaining n odd Sierpinski / Riesel (which can be tranlated as base 4, k even, and doubling the Gallot values : 190566 for k*4^n+1, 345354 for k*4^n-1) I began to explore the problem, by eliminating all k's yielding a prime for n < 4096, eliminating the perfect square k values for Riesel, eliminating the MOB that are redondant, etc... Finally, there were 42 k values remaining for +1, 114 for -1, and after sieving rapidly with NewPGen, and LLRing up to ~32K, I have now 21 values remaining for +1 and 37 values for -1. I would be happy to know your opinion about all that... Regards, Jean[/quote] Jean, Is there any reason that we are not testing even k's (i.e. multiples of the base) with these conjectures? If a k is even but is not divisible by 4, it yields a different set of factors and prime than any other odd k. I am testing the Sierp odd-n conjecture of k=95283. Can you tell me how you arrived at 21 k-values remaining at n=32K? I have now tested up to n=56K. I just now finished sieving up to n=200K and am starting LLRing now. At n=56K, I show 22 odd k's and 8 even k's remaining that are not redundant with other k's remaining; for a total of 30 k's. At n=32K, I showed 26 odd k's and 9 even k's remaining; for a total of 35 k's. I checked the top-5000 site for previous smaller primes and there were none for these k's so I wonder why you have less k's remaining than me. Here are the k's that I show remaining at n=32K, both odd and even, and primes that I found for n=32K-56K for the Sierp odd-n conjecture: [code] k comments/prime 2943 9267 17937 prime n=53927 24693 26613 29322 even 32247 35787 prime n=36639 37953 38463 39297 43398 even 46623 46902 even 47598 even 50433 53133 60357 60963 61137 61158 even; prime n=48593 62307 prime n=44559 67542 even 67758 even 70467 75183 prime n=35481 78753 80463 83418 even 84363 85287 85434 even 91437 93477 93663 [/code] Thanks, Gary |

[QUOTE=gd_barnes;122833]Jean,
Is there any reason that we are not testing even k's (i.e. multiples of the base) with these conjectures? If a k is even but is not divisible by 4, it yields a different set of factors and prime than any other odd k. I am testing the Sierp odd-n conjecture of k=95283. Can you tell me how you arrived at 21 k-values remaining at n=32K? I have now tested up to n=56K. I just now finished sieving up to n=200K and am starting LLRing now. At n=56K, I show 22 odd k's and 8 even k's remaining that are not redundant with other k's remaining; for a total of 30 k's. At n=32K, I showed 26 odd k's and 9 even k's remaining; for a total of 35 k's. I checked the top-5000 site for previous smaller primes and there were none for these k's so I wonder why you have less k's remaining than me. Here are the k's that I show remaining at n=32K, both odd and even, and primes that I found for n=32K-56K for the Sierp odd-n conjecture: [code] k comments/prime 2943 9267 17937 prime n=53927 24693 26613 29322 even 32247 35787 prime n=36639 37953 38463 39297 43398 even 46623 46902 even 47598 even 50433 53133 60357 60963 61137 61158 even; prime n=48593 62307 prime n=44559 67542 even 67758 even 70467 75183 prime n=35481 78753 80463 83418 even 84363 85287 85434 even 91437 93477 93663 [/code] Thanks, Gary[/QUOTE] In the definitions of these four conjectures, the k multipliers must be odd! For example : 46902*2^n+1 is the same as 23451*2^(n+1)=+1 and if n is odd, n+1 is even, so, you are testing an even exponents candidate! So, the 8 even k's remaining are relevant to the even n conjecture, and not to the odd n one... Also, I tested k = 46623 up to n = 79553 and found a prime, so me are almost matching now... I am terminating to gather my results, and will send them to this thread as soon as possible. Regards, Jean |

[quote=Jean Penné;122841]In the definitions of these four conjectures, the k multipliers must be odd!
For example : 46902*2^n+1 is the same as 23451*2^(n+1)=+1 and if n is odd, n+1 is even, so, you are testing an even exponents candidate! So, the 8 even k's remaining are relevant to the even n conjecture, and not to the odd n one... Also, I tested k = 46623 up to n = 79553 and found a prime, so me are almost matching now... I am terminating to gather my results, and will send them to this thread as soon as possible. Regards, Jean[/quote] Well, DUH!! :blush: I will remove the even k's that obviously go with the even-n conjecture from my LLRing. :rolleyes: Gary |

I have now run the Sierp odd-n conjectures up to n=115K and will be continuing on to n=200K sometime next week after completing some sieving for conjectures team drive #1 and a couple of other things. Below are the k's left at n=32K with primes found for n=32K-115K.
I decided to leave the even k's in because in effect it is testing the even conjecture for all k < 95282/2=47641 and I had already sieved them. That should save a lot of effort on that side. [code] k comments/prime 2943 prime n=108041 9267 17937 prime n=53927 24693 26613 prime n=89749 29322 even; prime n=91367 32247 35787 prime n=36639 37953 38463 prime n=58753 39297 43398 even; prime n=72873 46623 prime n=79553 46902 even 47598 even; prime n=105899 50433 53133 60357 60963 prime n=73409 61137 61158 even; prime n=48593 62307 prime n=44559 67542 even 67758 even 70467 75183 prime n=35481 78753 prime n=63761 80463 83418 even; prime n=80593 84363 85287 85434 even 91437 93477 prime n=63251 93663 prime n=82317 [/code] Total of 14 odd k's and 4 even k's remaining. So...based on this effort by itself, here are the statuses of the base 2 Sierp odd-n and even-n cojectures: Odd-n: 14 k's remaining at n=115K from odd k's above. k's remaining: 9267 24693 32247 37953 39297 50433 53133 60357 61137 70467 80463 84363 85287 91437 Even-n: 47641<k<66741: still needs to be tested. k<=47641: 4 k's remaining at n=115K from even k's above. k's remaining converted to odd-k: 23451 33771 33879 42717 Edit: I just now realized that it was already stated that only k=23451 and 60849 are remaining on the even-n side as a result of the Sierp base 4 project. OK, NEXT time I'll remove the even k's from my testing. Ergh! Gary |

Gathered results for k*2^n+1, n oddGary,
I am gathering my results about the four conjectures, which requires a lot of work... So, we will be able to compare with your results! Here for +1 and n odd : There are 23 remaining candidates, and 42 primes found. 1) Remaining : [CODE] k 2k tested up to (n-1 base 2) 9267 18534 1967862 32247 64494 1770506 37953 75906 33448 38463 76926 34320 39297 78594 35166 50433 100866 46076 53133 106266 87428 56643 113286 33348 60357 120714 46166 60963 121926 32912 61137 122274 33150 62307 124614 35342 70467 140934 46358 75183 150366 32840 78153 156306 32976 78483 156966 33096 78753 157506 55640 80463 160926 35660 84363 168726 35008 85287 170574 33106 91437 182874 33034 93477 186954 34846 93663 187326 67844 [/CODE] 2) Primes : [CODE]Normalized As discovered k n 93 20917 186*2^20916+1 is prime! Time: 786.013 ms. 2943 108041 5886*2^108040+1 is prime! by Jean Penné 06/02/05, 09:28AM 5193 4277 10386*2^4276+1 is prime! Time: 59.124 ms. 5703 5149 11406*2^5148+1 is prime! Time: 99.286 ms. 5823 8105 11646*2^8104+1 is prime! Time: 384.321 ms. 6807 4415 13614*2^4414+1 is prime! Time: 64.834 ms. 6843 14753 13686*2^14752+1 is prime! Time: 935.778 ms. 7233 4277 14466*2^4276+1 is prime! Time: 58.406 ms. 9777 18975 19554*2^18974+1 is prime! Time: 791.931 ms. 10923 6801 21846*2^6800+1 is prime! Time: 177.623 ms. 14397 4347 28794*2^4346+1 is prime! Time: 59.056 ms. 16917 12799 33834*2^12798+1 is prime! Time: 829.050 ms. 17457 29563 34914*2^29562+1 is prime! by tcadigan 28/01/05, 07:13AM 17937 53927 35874*2^53926+1 is prime! by tcadigan 28/01/05, 06:41AM 20997 8191 41994*2^8190+1 is prime! Time: 384.323 ms. 22653 28969 45306*2^28968+1 is prime! by Mark 27/01/05, 11:29PM 24693 357417 49386*2^357416+1 is Prime! by Footmaster 25/05/05, 08:44AM 25083 24981 50166*2^24980+1 is prime! by Mark 27/01/05, 11:29PM 25917 9671 51834*2^9670+1 is prime! Time: 447.572 ms. 26613 89749 53226*2^89748+1 is prime! Time: 25.000 sec. 30933 4433 61866*2^4432+1 is prime! Time: 59.831 ms. 35787 36639 71574*2^36638+1 is prime! Time: 6.016 sec. 40857 5383 81714*2^5382+1 is prime! Time: 154.925 ms. 42993 16165 85986*2^16164+1 is prime! Time: 1.223 sec. 43167 9795 86334*2^9794+1 is prime! Time: 403.492 ms. 46623 79553 93246*2^79552+1 is prime! Time: 28.184 sec. 49563 5813 99126*2^5812+1 is prime! Time: 186.733 ms. 60273 7421 120546*2^7420+1 is prime! Time: 218.279 ms. 63357 4211 126714*2^4210+1 is prime! Time: 136.667 ms. 65223 4189 130446*2^4188+1 is prime! Time: 136.782 ms. 65253 10301 130506*2^10300+1 is prime! Time: 427.195 ms. 67917 13079 135834*2^13078+1 is prime! Time: 644.241 ms. 69963 5205 139926*2^5204+1 is prime! Time: 152.156 ms. 72537 15771 145074*2^15770+1 is prime! Time: 1.199 sec. 73023 17965 146046*2^17964+1 is prime! Time: 1.349 sec. 78543 10089 157086*2^10088+1 is prime! Time: 419.114 ms. 80517 5423 161034*2^5422+1 is prime! Time: 154.232 ms. 81147 17615 162294*2^17614+1 is prime! Time: 1.331 sec. 82197 5079 164394*2^5078+1 is prime! Time: 149.421 ms. 88863 9825 177726*2^9824+1 is prime! Time: 405.011 ms. 91383 15333 182766*2^15332+1 is prime! Time: 1.173 sec. 93033 30473 186066*2^30472+1 is prime! Time: 5.037 sec. [/CODE] Note : The name of the discoverer is shown only for primes found by the Sierpinski base 4 project. Regards, Jean |

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