20171125, 22:36  #12 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
9,127 Posts 
If anyone was seriously interested in finding a PRP larger that the largest known prime, then consider this.
There are no numbers that can be tested faster than Mersenne's (within a fraction of a % of a %, that is your "less iterations to run for the cofactor" which is negligible if you think for a minute). To be even remotely competitive with GIMPS in terms of size and plausibility, the candidates have to have three properties: (a) as fast to test as a Mersenne number. (b) as probable to find as a Mersenne number (that means sieved just as deep, that means factors are restricted)  this is a must. If your project can sieve to 36 bits and GIMPS to 72, then you already lost. (c) finally, the project has a compute throughput as large as GIMPS  but for far less glory. Conditions (a+b) will mean that candidates have to be Mersenne cofactors, Wagstaff's, GFN cofactors or some cyclotomic cofactors (or straight with power not 3smooth). GaussianMersennes and EisensteinMersennes (and their cofactors) are not contenders (they fail condition (a), though they do pass condition (b))  failing condition (a) by a factor of 2 is as bad as failing condition (b) by a factor of 2  in both cases in order to simply keeping up with GIMPS you have to have 2x more power. But then  why? You could just as well work with GIMPS even if you consider getting half of credit with GIMPS getting the other half. 
20171125, 23:37  #13  
"Rashid Naimi"
Oct 2015
Remote to Here/There
7·271 Posts 
Thank you very much for the detailed analysis Batalov, much appreciated.
On a somewhat related benchmark Quote:
https://archive.lib.msu.edu/crcmath/...ath/c/c060.htm Last fiddled with by a1call on 20171125 at 23:38 

20171126, 14:35  #14  
Feb 2017
Nowhere
6632_{8} Posts 
Quote:
U3 = PQR = (6M + 1)(12M+1)(18M + 1) in which the prime factors P, Q, and R each have 1025 decimal digits. The post also refers to Carmichael numbers with "millions of digits." Carmichael number record 

20171126, 16:48  #15 
Feb 2017
Nowhere
2·1,741 Posts 
A bit more digging turned up a 1998 post by Dubner with some larger examples of the form
(6M + 1)(12M + 1)(18M + 1): 3component Carmichael Numbers There was an error in a table, which was the subject of a Correction The smallest Carmichael number of this form, 7*13*19, is 1729, which is the smallest Carmichael number whose prime factors form a finite arithmetic progression, as well as being (as noted in the hardyRamanujan "taxicab number" anecdote) the smallest number that is the sum of two cubes in two different ways (10^3 + 9^3 and 1^3 + 12^3). Also, a 1996 paper with an algorithm for producing really large Carmichael numbers: A NEW ALGORITHM FOR CONSTRUCTING LARGE CARMICHAEL NUMBERS 
20171126, 19:45  #16 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7·271 Posts 
Thank you very much for the corrections and extra research DS.
You are a very valuable member of this board. 
20171126, 19:51  #17 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7×271 Posts 
What is the largest PRP range that can be found using Fermat's Little Theorem (base 2) on Pari GP running on a custom pc with x GBs of memory?
Another way of asking the same question is: What is the largest practical p for which 2^(p 1) can be calculated? (Thank you in advance)^2 Last fiddled with by a1call on 20171126 at 19:54 
20171126, 20:01  #18  
Aug 2006
3×5^{2}×79 Posts 
Quote:
I love PARI/GP but this is like pounding nails with a Swiss Army knife. 

20171126, 21:17  #19  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
9,127 Posts 
Quote:
You have a better chance to be struck by a lightning than finding a PRP with > 10M decimal digits. Do you feel particularly lucky? You can run PRPs on the >100,000,000digit Mersenne cofactors (candidates that had a factor or two found). In a few CPU years you will get the result. Take 2^3322000071 / 272240898059289853159 and run the PRP test. It is trivial: Code:
# worktodo.txt PRP=1,2,332200007,1,"272240898059289853159" You can take even larger cofactors. 

20171126, 22:36  #20 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7·271 Posts 
Point well taken. I wasn't considering the low probability of prime/PRP in the high ranges.
But then again I am more interested in understanding the different mechanics of the concept rather actually attempting to find a large PRP. It's good to know the upper limits of what arbitraryprecision calculators can work at then consider the astronomically low possibility of finding a candidate. 
20171126, 22:41  #21  
"Rashid Naimi"
Oct 2015
Remote to Here/There
1897_{10} Posts 
Quote:
http://www.periodictable.com/Samples/026.55/s13.JPG You know what they say: You can't teach an old d d d distinguished gentleman new tricks. PFGW, at the moment is nothing but an acronym to me. I will have to research it. Thank you. 

20171126, 23:33  #22  
Sep 2003
2^{2}×3×5×43 Posts 
Quote:
If you're feeling less ambitious, then use 150 for firsttime PRP tests or 152 for worldrecord PRP tests. These two choices currently amount to the same thing, at least until the next Mersenne prime is found. It goes without saying that you should be running the latest version, 29.4b5 Last fiddled with by GP2 on 20171126 at 23:34 

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