20190507, 16:44  #12 
Sep 2002
Database er0rr
19·173 Posts 

20190507, 19:01  #13 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
13·19·23 Posts 

20190507, 19:25  #14 
Sep 2002
Database er0rr
6327_{8} Posts 
A little more on the subject... A proof that there are infinitely many pseudoprimes base a

20190509, 07:54  #15 
Romulan Interpreter
Jun 2011
Thailand
2×5×859 Posts 
And there may be none at all. Something similar to Pepin test for Fermats may go also for Mersennes, but this is still an open question (as opposite to a "conjecture", where we can "guess" or "heuristically prove" one way or the other, here we have no clue). My belief is that a mersenne number which is 3PRP is also prime, and this also applies to mersenne factors, if we restrict the exponent to be prime. But of course, I have no clue how could we prove that. This falls in the same category as the infinitude of Wieferich primes, or the mersenne numbers with prime exponents being square free  we have no guess, in either direction.
Last fiddled with by LaurV on 20190509 at 07:58 
20190509, 15:35  #16 
Sep 2009
43^{2} Posts 
Has anyone checked for a Mersenne number that's a Lucas pseudoprime? Any such would also be a BPSW pseudoprime which would be an interesting discovery. Or can it be proved there are none such?
Chris 
20190509, 16:01  #17 
Feb 2017
Nowhere
2·3·547 Posts 

20190510, 00:19  #18  
"Jeppe"
Jan 2016
Denmark
176_{8} Posts 
Quote:
But as you may have guessed, 10974881 is not a 5PRP. In fact: 10974881 = 1913 * 5737 Can someone come up with a conjecture on the number of counterexamples of this type? /JeppeSN 

20190510, 01:01  #19  
"Rashid Naimi"
Oct 2015
Out of my Body
31×59 Posts 
Quote:


20190510, 03:42  #20  
Romulan Interpreter
Jun 2011
Thailand
2·5·859 Posts 
Quote:
Never managed to do any tests, it was some small light blinking in a corner of my mind, and bothering me regularly from time to time, but I never invested the time to check. Now I am very glad you found that! It settles the blinking light... hehe It is a pity that in this case we still can't declare the "probable fully factored" mersennes that GP2 (and others) are putting a lot of effort into, to be "fully factored for sure" ... Last fiddled with by LaurV on 20190510 at 03:48 Reason: spaces 

20190510, 04:39  #21  
Sep 2003
A13_{16} Posts 
Quote:
Edit: actually, FactorDB only stores PRPs for Mersenne cofactors for exponents up to about 500k. So for the larger ones, there is no record of testing them to any base other than 3. You could do it with PFGW. In any case Paul Underwood posts his Lucas PRP results to the "disbelievers" thread. Last fiddled with by GP2 on 20190510 at 04:49 

20190510, 05:09  #22 
"Jeppe"
Jan 2016
Denmark
2·3^{2}·7 Posts 

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