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

- - **Bases 33-100 reservations/statuses/primes**
(*https://www.mersenneforum.org/showthread.php?t=10475*)

Riesel base 451 Attachment(s)
Hi everyone, here is my base 45 effort. The Riesel conjecture is 22564.
I've taken this to n = 10,000. while ignoring odd k and (k mod 11) = 1. As 1080 = 24*35 and 16740 = 372*45 I've left hem out. This leaves the 22 following k: 24 372 1264 1312 2500 2804 4210 4484 5128 6094 6372 7246 10096 10518 12950 13456 13548 15432 17918 19252 20654 21274 There are 2 squares remaining, 2500 and 13456, but that is just coincidence I think. There are many square k's that do have a prime. Also, 24 = 6*2*2. Several of the conjectures remove k = 6*square, but I don't understand why. How can I check if it can be removed here also? The top ten of primes: 13546 9069 17734 8019 19102 7368 14324 7281 9938 7240 4628 7209 4622 7116 6554 6462 2230 5892 7750 4586 I've attached and doublechecked all the primes that I found. Willem. |

[quote=Siemelink;138726]
There are 2 squares remaining, 2500 and 13456, but that is just coincidence I think. There are many square k's that do have a prime. Also, 24 = 6*2*2. Several of the conjectures remove k = 6*square, but I don't understand why. How can I check if it can be removed here also? Willem.[/quote] It's not always very easy. The way I do it is to look for patterns in the factors of the various n-values for specific k-values. If there are algebraic factors, it's most common for them to be in a pattern of f*(f+2), i.e.: 11*13 179*181 etc. In other cases there may be a consistent steady increase in the differences of their factors, which is especially tricky to find but indicates the existence of algebraic factors. That's what I ran into on base 24. For your 3 cases here, you have: k=24: n-value : factors 1 : 13*83 2 : 23*2113 3 : 17*103*1249 4 : 23*163*26251 9 : 2843*6387736694293 Analysis: For n=3 & 4, multiplying the 2 lower prime factors together does not come close to the higher prime factor so little chance of algebraic factors. For n=9, the large lowest prime factor that bears no relation to the other prime factor means that there is unlikely to be a pattern to the occurrences of large prime factors so there must be a prime at some point. k=2500: n-value : factors 1 : 19*31*191 2 : 13*173*2251 3 : 89*2559691 4 : 19*73^2*103*983 9 : 9439*4280051*46824991 For n=9 same explanation as k=24. k=13456: n-value : factors 1 : 269*2251 2 : 17*23*227*307 3 : 31*39554129 4 : 7*23*467*503*1459 7 : 3319*1514943103721 For n=7 same explanation as k=24. The prime factors for n=9, n=9, and n=7 respectively make it clear to me that these k-values should all yield primes at some point so you are correct to include them as remaining. The higher-math folks may be able to chime in and answer why there are an abnormally large # of k's that are perfect squares that end up remaining even though they don't have known algebraic factors for most bases. IMHO, it's because there ARE algebraic factors for a subset of the universe of n-values on them but not for all of the n-values. Hence they are frequently lower weight than the other k's but NOT zero weight and so should eventually yield a prime. Gary |

base 42 Riesel1 Attachment(s)
Hi everyone,
here is my work on the base 42 Riesel conjecture 15137. The covering set is {5, 42, 353}. After taking n to 10,000 there were 72 k left. I removed 2058 = 49 * 42 as 49 is still in the list of k. There are also 5 squares in the list (49, 1369, 2304, 3721 and 10201), but they give no obvious deductions. This leaves: [code] 49 386 603 1049 1160 1426 1633 1678 2304 2464 2538 2753 3428 3734 4299 4903 5118 5417 5677 5820 5899 5978 6333 6623 6664 6836 6838 6964 7016 7051 7309 7489 7614 7658 7698 7913 8297 8341 8384 8453 8524 9029 9201 9418 9633 9848 10026 10114 10276 10663 10923 11052 11267 11781 11911 11996 12039 12125 12127 12151 12213 12598 13288 13329 13347 13425 13632 13757 13898 14576 15024 [/code] Enjoy, Willem. |

[quote=Siemelink;139767]Hi everyone,
here is my work on the base 42 Riesel conjecture 15137. The covering set is {5, 42, 353}. After taking n to 10,000 there were 72 k left. I removed 2058 = 49 * 42 as 49 is still in the list of k. There are also 5 squares in the list (49, 1369, 2304, 3721 and 10201), but they give no obvious deductions. Enjoy, Willem.[/quote] Looks good. Nice work. Actually only 2 of your squares are remaining: k=49 and k=2304. As you showed in your list, k=1369 has a prime at n=7577, k=3721 has a prime at n=4611, and k=10201 has a prime at n=2129. Gary |

Riesel base 371 Attachment(s)
Hi there Gary, thanks for clarifying my muddled statement.
here is the next one, base 37. The conjecture is 7772, with set = {5, 19, 137}. At n = 10,000 there are 30 k remaining: [code] 284 498 522 590 672 816 1008 1578 1614 1842 1958 2148 2606 2640 3336 3480 3972 4356 4428 4542 4806 5262 5376 5910 5946 6288 6752 6792 7088 7352 7466 [/code] I've attached the primes found. Here is the list of the highest primes: 7058 8314 1334 7883 5156 7797 6480 7763 554 7472 7124 6396 474 3952 998 3572 912 3394 1956 3250 Willem. |

Riesel base 351 Attachment(s)
Hi everyone,
I've double checked my work on the Riesel 35 Conjecture. The value is 287860. I've taken the k's to n=200 with PFGw. Then I sorted out the multiples of 35 and the squares. ThenI took the remainder to n=5000 I've attached the 1559 k's that I had left. The top ten primes list is this: 65216 4986 248264 4980 104690 4978 126050 4978 286652 4976 129052 4975 229454 4974 48772 4965 169448 4964 7874 4962 All the primes are in a file that zips to 350k, that doesn't fit on the forum. The can be sent if so desired. Willem. |

[quote=Siemelink;144576]Hi everyone,
I've double checked my work on the Riesel 35 Conjecture. The value is 287860. I've taken the k's to n=200 with PFGw. Then I sorted out the multiples of 35 and the squares. ThenI took the remainder to n=5000 I've attached the 1559 k's that I had left. The top ten primes list is this: 65216 4986 248264 4980 104690 4978 126050 4978 286652 4976 129052 4975 229454 4974 48772 4965 169448 4964 7874 4962 All the primes are in a file that zips to 350k, that doesn't fit on the forum. The can be sent if so desired. Willem.[/quote] Can you please send the primes to me at: gbarnes017 at gmail dot com Thanks, Gary |

i give base 35 a few days shot!
so far: sieved all 1559 (minus k with primes) upto p=402M for n=5k-100k checked upto n=5249 38 more primes found 6.5M candidates left sieving further! will mail primes when more available. |

reserved base 35 from n=5k-100k
so Siemelink can check another base :-) |

Rest assured that it'll take you quite some time to take it to 100k...
(Base 31, also not very prime (compared to base 3), now runs erm... 4 or 5 months or so :) ) |

[quote=michaf;144873]
Rest assured that it'll take you quite some time to take it to 100k... (Base 31, also not very prime (compared to base 3), now runs erm... 4 or 5 months or so :) ) [/quote] To clarify: Actually base 31 is very prime compared to most bases. It is much more prime than base 35 is. But of course nothing compares to base 3. So far, only base 7 comes close. I also suspect base 15 will be quite prime. It seems that all bases where b=2^q-1 are very prime as compared to their neighbors. I would expect many CPU years to get base 35 up to n=100K. Gary |

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