20160814, 13:40  #1  
Aug 2015
2·23 Posts 
Exponent fully factored whilst only 74% known
Can someone explain why M397 is labeled as fully factored, when the details are:
Quote:
I also note that there are no PRPs attached to M397, which I would have expected with the mention of a probablyprime. 

20160814, 14:50  #2  
Jun 2003
2×3^{2}×17^{2} Posts 
Quote:
1) Because it is. 2) Lazy programmer. In this particular case, all the factors are small and proven prime, but in the general case, a fullyfactored exponent will result from a bunch of small factors and a large PRP cofactor. We're confident that the PRP cofactor is prime, but it will not be mathematically proven. There is no separate logic in the website to handle when the last factor is proven vs PRP. 

20160814, 14:51  #3  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2·23·137 Posts 
Quote:
It is a real prime BTW 

20160814, 18:09  #4 
Sep 2003
5×11×47 Posts 
By convention, if an exponent is fully factored, the largest factor is omitted from the database (in both mersenne.org and mersenne.ca).
Storing it would be redundant, since it easily calculated by dividing the Mersenne number itself by all of its other factors. And in most cases (although not here), it is many orders of magnitude larger than the next largest factor, so there would be enormous storage costs, for example: M5240707 = 75392810903 * (a probable prime with 1.5776 million digits) Note that the 273 fully factored Mersenne exponents up to and including M63703 really are fully factored, with primality certificates calculated for the remaining cofactor using programs like Primo, whereas all larger "fully factored" Mersenne exponents (of which 30 are known) are only "probably fully factored". Last fiddled with by GP2 on 20160814 at 18:15 
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