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Old 2022-06-06, 13:13   #1
RobSartain
 
Jun 2022

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Default 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 1999-12-01, then a factor was found, then it was proven composite again on 2017-12-30 albeit with different residue values, then another factor was found, then it was proven composite a third time on 2022-06-03 when I certified Loic's PRP work.

Why was it tested a total of three times? The different residue values were each double-checked, 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
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Old 2022-06-06, 13:43   #2
Uncwilly
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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.
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Old 2022-06-06, 15:40   #3
ATH
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Yes, this is the initial test + double check on M3936727 = 23936727-1
Code:
LL	
Status			Date		User		Residue			Shift
Verified (Factored)	1998-01-26	Soh Kok Hong	7D98BB85E98906AA	0
Verified (Factored)	1999-12-01	Rob Simons	7D98BB85E98906AA	2768285
Then the 1st factor was found and we check if the remaining cofactor M3936727 / 1416543718452177641 is prime or composite. This is not part of the main GIMPS project to find Mersenne Primes, but a side project to find as many factors as possible and check if we found them all for each exponent.

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)	2022-06-03	Loic		70D58841F6D82E77	2		3198615	5	3
Verified (Factored)	2017-12-30	Oliver Kruse	81659F6303C264C5	1		1531080	1	3
Verified (Factored)	2017-09-20	kkmrkkblmbrbk	81659F6303C264C5	1		0	1	3
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Old 2022-06-06, 17:00   #4
RobSartain
 
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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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