20220606, 13:13  #1 
Jun 2022
2 Posts 
Question about history of M3936727
Hi, all,
Can you help me understand the history of M3936727, please? https://www.mersenne.org/report_expo...3936727&full=1 As I read it, the exponent was proven composite on 19991201, then a factor was found, then it was proven composite again on 20171230 albeit with different residue values, then another factor was found, then it was proven composite a third time on 20220603 when I certified Loic's PRP work. Why was it tested a total of three times? The different residue values were each doublechecked, so I assume they're accurate, but is there significance to them differing? No biggie; I'm just curious how the PrimeNet assignment process worked in this case. Thanks for any insight! Rob 
20220606, 13:43  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{2}·7·389 Posts 
The initial test was an LL test. The residue it gives does not relate to the PRP residue. In 2017 to verify the PRP run on the the cofactor (the part that remains after dealing with the known factor[s]), we had to rerun the test with a different shift. Now we can do a cert on the PRP run. If you not the Type column that new 1 was a different type. And this was run after another factor was found. For numbers like this we run a new PRP to check the newly discovered cofactor to see if it is prime. If it is, we know that we have fully factored the number.

20220606, 15:40  #3 
Einyen
Dec 2003
Denmark
2·17·101 Posts 
Yes, this is the initial test + double check on M3936727 = 2^{3936727}1
Code:
LL Status Date User Residue Shift Verified (Factored) 19980126 Soh Kok Hong 7D98BB85E98906AA 0 Verified (Factored) 19991201 Rob Simons 7D98BB85E98906AA 2768285 The first 2 PRP Cofactor tests was in 2017 before PRP Certification / Proof and I think back then when PRP CF was just implemented only "Type 1" test was available. Then 2nd factor was found and now we test the remaining cofactor: M3936727 / (1416543718452177641*2542680988412771022521) so it gives a different residue with 2 factors, and this time with a "Type 5" test, which will give a different residue from Type 1 even if the number of factors was the same. The advantage of Type 5 is if a 3rd factor is found and Type 5 is run on the remaining cofactor, it will give the same residue as with 2 factors, so only 1 test is needed (and no PRP Certification needed). This advantaged with Type 5 is not that important now that we have PRP Certifications. Code:
PRP Cofactor Status Date User Residue Number of known factors Shift Type Base Verified (Factored) 20220603 Loic 70D58841F6D82E77 2 3198615 5 3 Verified (Factored) 20171230 Oliver Kruse 81659F6303C264C5 1 1531080 1 3 Verified (Factored) 20170920 kkmrkkblmbrbk 81659F6303C264C5 1 0 1 3 
20220606, 17:00  #4 
Jun 2022
2_{8} Posts 
Thanks very much, Uncwilly, ATH. I've been a GIMPS contributor since around 1997ish, but I didn't appreciate those nuances. Thanks again for replying!

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