20211120, 23:51  #1 
Aug 2020
2·5 Posts 
Primality Certificates.
Are there any reliable Primality Certificates for Mersenne Numbers that are prime that cannot be faked?

20211121, 12:03  #2  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
17F4_{16} Posts 
Quote:
To my knowledge no Mersenne prime discovery has yet been made by running PRP first, giving a PRPP result, and then confirming with LL tests, as would occur for the next one. So consequently none have been found by PRP and proof generation. M82589933 is the only one that might have happened on, since the first PRP tests were for ~77.5M and above, so excluding the next lower M77232917 and lower. See also the last several entries on https://www.mersenneforum.org/showpo...46&postcount=6 for some related links. 

20211121, 13:32  #3 
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
OP wants something different. A primality certificate for Mersenne primes, so a (very) fast proof when 2^p1 is prime (where p is an odd prime).
Let r=2+sqrt[3] and calculate v=r^(2^(p2)) in Z[N,sqrt[3]], where N=2^p1. Lucas Lehmer test says (in an equivalent form) that N is prime iff v=k*sqrt[3]. The advantage of this form is that you can use error checking and giving certificates as like for 3^(2^p) mod N calculation without any probable prime issues, giving an answer to your question. It is hard to fake this. But why are we using the prp form? Because the calculation of v is at least two times slower than a standard LL or prp test with error checking. 
20211121, 15:46  #4 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{2}×3×7×73 Posts 
This paper states the Pepin test for Fermats, and the LucasLehmer for Mersennes, provide certificates.
https://www.ams.org/journals/mcom/19...08661174.pdf 
20211121, 16:55  #5  
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
Quote:


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