20140308, 23:44  #1  
Dec 2010
2×37 Posts 
P1 factoring question
From "The Math" section of GIMPS:
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I would think E should be the product of all primes less than B1, all squares of primes less than the the square root of B1, all cubes of primes less than the cubic root of B1, etc. Or am I way off here, missing something about the math? When reading about P1 factoring, K! was used instead which contains more than a sufficient number of powers of small primes. If I am catching a missed opportunity here, then P1 factoring for Mersenne numbers could be very substantially improved on using additional powers of the small primes. After all, numerous k values of real factors are divisible by 4, 8, 16, 32, 64, 128 or 9, 27, 81, or 25, 125, or 49, 121, 169, etc. Last fiddled with by siegert81 on 20140308 at 23:53 

20140309, 01:06  #2  
"Richard B. Woods"
Aug 2002
Wisconsin USA
17014_{8} Posts 
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You've found a documentation bug, or at least a nonclarity. Quote:
But in fact, the few missed factors that are occasionally discovered have been missed because of other problems, not because of insufficient primepowering in Prime95's implementation of P1. Last fiddled with by cheesehead on 20140309 at 01:14 

20140309, 05:04  #3 
Romulan Interpreter
"name field"
Jun 2011
Thailand
2×17×293 Posts 
Saying it differently, the described method guarantees that it searches all the "searching space" under B1, i.e. it will guarantee to find a factor q if q1 is B1powersmooth. Only talking about stage 1 here. Stage 2 is a different fish. Of course, you can use higher powers if you like, that would be a "shot in the dark", trying to find some factors "out of the searching space". It will be called an "extension" to the algorithm, and that is what BrentSuyama extension is doing. You can do any type of "extensions" you like, but not all of them are worth a try, you can work in vain ages without finding a factor. Remark that you do not "extend" B1, by using larger powers, unless you do not add new primes to the exponent. Take a small example and try to do it practically.

20140309, 12:38  #4  
Dec 2010
2×37 Posts 
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