20210206, 14:50  #1 
Nov 2020
Massachusetts, USA
2×7 Posts 
No factors below 77 bits  Are we sure?
There seem to be a lot of exponents in the 109M range for which the first logged event was "No factors from 2^63 to 2^64" (example: https://www.mersenne.org/report_expo...exp_hi=&full=1) and although they have been trialfactored to 2^76 or even 2^77 without finding a factor, it appears that no one has tested the range below 2^63. An even crazier example is this: https://www.mersenne.org/report_expo...exp_hi=&full=1. Notice that unlike the first example, this example does not have bounds B1 nor B2. User "GPU Factoring" started a TF back in May, with most recent update 20210130, but has never finished it. Again, it is known that no factors exist between 2^63 and 2^74, but for all we know, maybe there is a factor between 2^1 and 2^63 that has been hiding very well and evaded all previous factoring attempts. My question is: Should we take the "No factors below 2^74" with a grain of salt? Is there a chance that it may actually be incorrect? It wouldn't even be the first time, either: see https://www.mersenne.org/report_expo...4503921&full=1. In September 2015, it was TF'd to 2^67, so the top of the exponent's entry said "No factors below 2^67" for a while, until six months later, when Mark Rose found the factor 65655858100148241169, which is less than 2^66.

20210206, 14:55  #2 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
1011111100111_{2} Posts 
For factoring: No, we are not sure. It is mostly based upon trust in what people report.
There is the possibility to detect anomalies in the rate of factor finding when looking each users results. Too few factors found might indicate a hardware issue or fraud. So far, AFAIAA, no fraud has been found, but hardware issues have been identified. Last fiddled with by retina on 20210206 at 14:56 
20210206, 16:17  #3 
Einyen
Dec 2003
Denmark
3×5^{2}×41 Posts 
The missing trial factoring between 0 and 2^63 or 2^64 is supposed to be a primenet bug, that it is not showing, but it was done initially.
I'm not sure what caused the bug, it could not have been the conversion from primenet v4 to v5 I think, since the range 79.3M1000M was not opened up until v5 ? Unless factoring results for 80+M was kept somewhere else like mersenne.ca and then imported to primenet v5 back then. I did test a range of like 1M exponents once, where I did all the missing 0 to 2^64, and I did not find any factors. But it was a tiny range compared to the whole 80M1000M. Prime95 / mprime could be set to do 0 to 2^64 automatically before any PRP or LL test just to be sure, since it goes so fast even on old computers. But that will not apply to all the gpuowl and CUDALucas tests. Last fiddled with by ATH on 20210206 at 16:21 
20210206, 16:24  #4 
Sep 2009
2^{2}·3·13^{2} Posts 
All factors of Mersenne numbers must be of form 2kp+1 where p is the exponent. So the lowest several bits are covered by p.
And the server itself has done checks over the bottom of the range. Which should cover up to 63 bits. And if a factor was missed by TF it could still be found by P1. So there are not likely to be many missed factors. Chris. 
20210206, 16:40  #5  
If I May
"Chris Halsall"
Sep 2002
Barbados
7×1,361 Posts 
Quote:
For GIMPS' purposes, not the end of the world. This is *not* GIMFS! 9) The FC/DC/et al process is authoritative of primality. Worst case scenario is more of this might be done than was optimal. Any nonnominal TF success rate of particular users is regularly reviewed. To the best of my knowledge, no intentional cheating has ever been found, although bad kit and/or code /has/ been from timetotime. 

20210206, 16:45  #6 
Apr 2010
Over the rainbow
43·59 Posts 
Some factor have been missed with pm1, but only from very early version of P95/mprime.
These can be found here : https://www.mersenne.ca/p1missed.php Around 2750 at this time Last fiddled with by firejuggler on 20210206 at 16:46 
20210206, 16:50  #7 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{5}×157 Posts 
See what's now the last paragraph of https://www.mersenneforum.org/showpo...6&postcount=2; begins "The absence of"

20210206, 17:20  #8 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
9488_{10} Posts 
To further give assurance to the data. A user known TJAOI (ranked 33 here) has been systematically going through exponents bit level by bit level doing TF and reporting factors found. Their efforts have shown that the data on PrimeNet has been usually correct, with no obvious cheating (that was no previously investigated). They continue to search for factors even if a factor has been found for an exponent. And since they are going bitwise without worrying about the extra effort it takes for smaller numbers to get to various bit levels vs higher numbers, they have provided a great check. They are now using methods other than TF for the lower ranges. Their TF software appears to be of their own design (thus providing for a good software check vs Prime95 and the various other programs that we are using.

20210206, 19:09  #9  
P90 years forever!
Aug 2002
Yeehaw, FL
3^{2}·823 Posts 
Quote:
However, P1 probably would have found such a small factor. If that fails TJAOI has looked for all factors below ~2^66 for all exponents. In conclusion, it is very highly unlikely that a tiny factor has been missed. 

20210206, 22:25  #10  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
41·229 Posts 
Quote:
What is important that there are no false negatives by design. Candidates are removed with 100% evidence that they are not pregnant. Primes will not slip through. (There will be a negligible amount of needless primality tests  if ever a factor was missed. This is very rare.) Compare to a very curious experimental design (implemented elsewhere on the web): 1. candidates are checked for factors 2. candidates are doublechecked for factors (!!) 3. if there is no quorum of 2, candidates are triplechecked for factors (!!) ... and repeat until quorum of 2 4. candidates are checked for primality ....once (!!) Homework: What could possibly be a problem with this design? 

20210214, 06:54  #11 
Aug 2020
1100001_{2} Posts 
Batalov, do you refer to Primegrid?
There's no real doublecheck anymore, but from what I read the new system provides the same level of safety. 
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