20210415, 16:09  #1 
Oct 2020
Terre Haute, IN
68_{10} Posts 
Question about the process for determining primes
What level of exponent will be checked (2^78, maybe?) for factors before it can be determined that in fact a number is prime? I ask that because currently one of my computers is doing a doublecheck on an eightdigit exponent that at the moment does not have any factors that have been determined, according to mersenne.ca.:
https://www.mersenne.ca/exponent/60884413 
20210415, 16:36  #2 
"Curtis"
Feb 2005
Riverside, CA
3×5×11×29 Posts 
Consider the number 143. How far do you have to check it for factors to determine if it's prime?
Either a composite number is a perfect square, or one of its factors is smaller than its square root. So, if you check all factors below a number's square root and find no factors, you have found a prime. But, as 143 shows, no lesser check is sufficient if you intend to determine primality by factorchecking alone. 
20210415, 19:13  #3  
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
61·157 Posts 
Quote:
https://www.mersenne.org/report_expo...0884413&full=1 74 bits is the reasonable level for the number to be factored to before switching to a primality test. It already had that done. It also had P1 done using B1 and B2 such that there was a 4.22% chance of finding a factor, it was done between the 73 and 74 bit runs, otherwise it would have had a 3.83% chance. 

20210416, 23:34  #4  
Oct 2020
Terre Haute, IN
44_{16} Posts 
Quote:
I come to this site a lot because I know I can find a lot of people here who are smarter than me, and I'm hoping it rubs off! 

20210417, 01:08  #5 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
22551_{8} Posts 
The process for looking for Mersenne Primes flows like this:
Mersenne.ca is largely an info site. It has some helper tools. It also is coordinating TF on exponents higher than the PrimeNet server is currently handling (and are so big that we should not worry about them for several decades/centuries. GPU72.com is a helper site that is designed to manage the TF (and P1) work in a more complex way and squeezing the optimum amount of work out of GPUs before turn the exponents over for primality testing.. 
20210417, 14:04  #6 
Romulan Interpreter
Jun 2011
Thailand
10010011100111_{2} Posts 
Very nice summary Unc.
To add to it, about half of the exponents are eliminated by TF and P1. For the rest, PRP will be run. If PRP turns out a probable prime, LL will be run to confirm it. This is the current scenario, the rest is history. Last fiddled with by LaurV on 20210417 at 14:05 
20210417, 15:43  #7 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·3^{3}·5·19 Posts 
To clarify on TF software:
Only mfaktc on NVIDIA gpus and mfakto on AMD gpus or perhaps OpenCL supporting IGPs are worthwhile in the mersenne.org exponent range. There are better uses for cpus: P1, DC, PRP. Mfactor and the slower factor5 are useful for preliminary work above the 4^321 range of mfaktx. It's more like 2/3 of prime exponents get eliminated by adequate TF or P1. As of Jan 27 2021: From 2 to 100M: ~10^8 natural numbers; ~94% of those are composite and eliminated as exponents; ~65.6% of the prime exponents were eliminated by finding factors with trial factoring or P1 factoring; that left ~1,983,000 for primality testing; 51 primes were found (~8.85 ppm of prime exponents) From 100M to ~332M: ~232,000,000 natural numbers; ~12,100,000 primes as possible exponents, a 94.8% reduction; ~60.6% were eliminated by trial factoring and P1 factoring by Jan 27 2021, with more remaining to do; expected number of primes to be found among them, ~3.1; that's 0.25 ppm of prime exponents; effort for a 100Mdigit primality test is ~12.4 times that of a 100M exponent's test From ~332M to 999,999,999: ~668,000,000 natural numbers; ~33,000,000 primes as possible exponents (95% reduction); ~19,000,000 already eliminated by Jan 27 2021 with trial factoring or P1 factoring (with much more left to do; >57% reduction so far) expected number of Mersenne primes to be found in this range ~2.83; that's 0.086 ppm chance per prime exponent; effort for a 999M exponent primality test is ~126. times that of a 100M exponent; probability per unit effort of finding a new Mersenne prime near 999M is 0.0077% that of the average in the range under 100M. Detailed tables summarizing the January 27 2021 state of http://hoegge.dk/mersenne/GIMPSstats.html are available here. Last fiddled with by kriesel on 20210417 at 15:44 
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