20220318, 22:28  #364 
Feb 2017
Nowhere
2^{3}×739 Posts 
Not one, but two correct answers to my exercise!
Well, at least a little bit of good mathematical thinking came of the latest shambles of a prediction for M52... 
20220318, 23:27  #365  
"Tucker Kao"
Jan 2020
Head Base M168202123
2^{4}×47 Posts 
Quote:
Quote:
In a (trine + 1) base, if the sum of the digits is trine, then the number is trine. What's the rule for (trine + 2) base? (3, 6, 11_{8}, 14_{8}, 17_{8}, 22_{8}, 25_{8}, 30_{8}) Last fiddled with by tuckerkao on 20220318 at 23:56 

20220319, 01:38  #366  
Jan 2021
California
670_{8} Posts 
Quote:
A simple way is to break the number into two digit pairs, if there's an odd number of digits, the first digit is by itself. Sum those numbers up, and the result will be the same mod 3 as the original number. You can repeat the process until you have a 2 digit number, then add twice the first digit to the 2nd digit, and that will also be the same mod 3 as the original number. Basically you are looking at the number in the base b^{2}, b^{2} will be congruent to 1 mod 3. 

20220319, 02:07  #367  
"Tucker Kao"
Jan 2020
Head Base M168202123
2^{4}×47 Posts 
Quote:
Dozenal divisibility of 5: Alternating sums of the numerical blocks of size 2. [Dozenal]25 * 5 = 101 Dozenal divisibility of 7: Alternating sums of the numerical blocks of size 3. [Dozenal]187 * 7 = 1001 Last fiddled with by tuckerkao on 20220319 at 02:19 

20220319, 04:48  #368  
Feb 2017
Nowhere
2^{3}×739 Posts 
Quote:
There should be a Forum rule against deliberately using gratuitously obscure lingo like "trine" when standard terminology (e.g. divisible by 3, congruent to 0 (mod 3)) exists. Note that b^k == (1)^k mod (b+1). Let d_{k} be the b^{k} baseb digit of n. Then n == d_{0}  d_{1} + ... (mod b+1), the alternating sum of baseb digits starting with the units digit. For the decimal base 10 = ten, the alternating digit sum determines whether n is divisible by 11. If b == 2 (mod 3) then 3(b+1), so this alternating digit sum is congruent to n (mod 3). 

20220319, 13:24  #369  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{2}·1,669 Posts 
Quote:


20220730, 07:56  #370  
"Tucker Kao"
Jan 2020
Head Base M168202123
2^{4}×47 Posts 
Quote:
Also, why M42643801 had the discovery date of Jun 4, 2009 not Apr 12, 2009? Last fiddled with by tuckerkao on 20220730 at 08:10 

20220730, 10:24  #371  
Einyen
Dec 2003
Denmark
3·1,117 Posts 
Quote:


20220730, 11:00  #372 
Einyen
Dec 2003
Denmark
3×1,117 Posts 
Here is the list using the agreed upon discovery dates, and today July 30th we reached the previous record gap of 1331 days, so tomorrow will be a new record This is not a record we want to beat.
Here is a "countup" timer since the discovery of M(82589933): https://www.timeanddate.com/countdow...3&font=cursive Code:
M(1398269) Nov 13 1996 M(2976221) Aug 24 1997 +284 days M(3021377) Jan 27 1998 +156 days M(6972593) Jun 1 1999 +490 days M(13466917) Nov 14 2001 +897 days M(20996011) Nov 17 2003 +733 days M(24036583) May 15 2004 +180 days M(25964951) Feb 18 2005 +279 days M(30402457) Dec 15 2005 +300 days M(32582657) Sep 4 2006 +263 days M(43112609) Aug 23 2008 +719 days M(37156667) Sep 6 2008 +14 days M(42643801) Jun 4 2009 +271 days M(57885161) Jan 25 2013 +1331 days M(74207281) Jan 7 2016 +1077 days M(77232917) Dec 26 2017 +719 days M(82589933) Dec 7 2018 +346 days  +8059 days (=503.7 days average) 
20220730, 15:50  #373  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{2}×1,669 Posts 
Quote:
I expect multiyear to decades intervals between Mp discoveries going forward, at least until one or more of the following occur to change the effort scaling, effort applied, or discovery rate: 1) quantum computing or graphene computing become readily available and widely employed in the search 2) we hit another lucky stretch with very closely spaced Mp 3) many times more participants deliberately join the project (or, mprime/prime95 service versions are implemented as standard installed apps or the idle loop in future releases of some major operating systems, which seems unlikely) The number of computer owners globally is about 10^{6} times as many as the number of GIMPS users seen active in the past 30 days. 4) some number theoretical advance is found, and significantly speeds the testing process further. I think the odds of this are low. Some parts of GIMPS are approaching known theoretical limits. (fft multiplication or squaring efficiency, nearly complete elimination of doublecheck effort by PRP/proof, and reduction in erroneous primality test by GEC) 5) ASICs such as Google TPUs become useful in the search and heavily utilized 6) some other transformative change not anticipated Ben Delo's contribution rate has declined. User count has declined. I hope to see at least one more Mp discovered in my lifetime. If one is located near 1.47576 x 82.6M ~122M, reaching and finding it may only take a few more years from now. I'd be happy to be shown unduly pessimistic in regard to time required to find Mp52* or Mp53*. Some very rough schedule estimates for discovering future Mp, assuming 1.47576:1 spacing and ~6M/year first test advance (optimistic compared to first half of 2022): Code:
Mp*# p~ year delta years from ~M(#1) 52 122M 2024 6 53 180M 2034 10 54 265M 2048 14 55 392M 2069 21 56 578M 2100 31 57 853M 2146 46 Last fiddled with by kriesel on 20220730 at 15:52 

20220730, 20:43  #374  
Einyen
Dec 2003
Denmark
3×1,117 Posts 
Quote:


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