Mersenne Digest Thursday, March 18 2004 Volume 01 : Number 1106
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Date: Thu, 12 Feb 2004 03:42:52 -0500
From: "Richard Woods" <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Factoring above 2^67
Emrecan Dogan wrote:
> One of my computers is assigned to run factoring tests
> continuously. I saw that my that computer's last two results
> were "no factor up to 2^67" . Then i checked the "cleared
> exponents list" on the primenet webpage. There were no trace of
> my exponents
That's because the Cleared Exponents Report includes only exponents
that have been "cleared" from being Mersenne prime candidates
because they've been proved to be composite. The two ways of
proving a Mersenne number composite are (1) find a factor, or (2)
complete a Lucas-Lehmer test with a nonzero residue. (Both
first-time L-L tests and doublechecks are included in the report.)
Exponents (such as your two) for which factoring has been attempted
but no factor has yet been found nor any L-L test completed are not
yet "cleared", so don't appear in the Cleared Exponents Report.
> but i saw many factored exponents with a factor bound to 2^101,
> 2^71, 2^68, 2^86 and so on.
Factoring assignments handed out by PrimeNet currently are only for
the so-called "trial factoring" method. Other factoring methods
(such as "P-1" and "ECM") exist and can be invoked by appropriate
commands in the worktodo.ini file even though PrimeNet doesn't yet
assign them separately. (P-1 factoring, if not previously performed
on the exponent, is now automatically performed at the beginning of
an L-L assignment).
However, when Prime95 was first written, the only factoring method
it incorporated was trial factoring, and most messages,
documentation and reports said just "factoring" without specifying
"trial factoring". When the P-1 and ECM factoring methods were
added to Prime95, the program's existing messages and reports
weren't modified to identify the specific factoring method in use.
When the Cleared Exponents Report shows a 101-bit factor, that's not
because trial factoring up to 2^101 was performed -- it's because
P-1 or ECM factoring found that factor. The P-1 and ECM methods
don't search for factors in the same way as trial factoring
searches. Their search bounds are not expressed as being "up to
2^86" (or whatever), and they are capable of finding factors much
larger than any feasible trial factoring run can find. The report
doesn't specify which factoring method was used to find a reported
factor, and it doesn't specify the search bounds used -- the bits
listed are the size of the factor, not the bound to which the
factor search was performed.
> What is the criteria that an exponent will be tested up to which
> power?
Although factoring could be performed to very high limits, the
probability of finding a factor does not increase as fast as the
time spent in searching for one. For each exponent, there is a
balance point below which it is most efficient (for maximizing GIMPS
throughput in determining whether Mersenne numbers are prime) to
search for a factor before L-L testing, but above which it is more
efficient (in terms of determining primality) to perform the L-L
test instead of continuing a factor search.
Currently, Prime95 uses the following default limits for trial
factoring:
Exponent range Trial factored to
- ----------------- -----------------
0- 1480000 2^56
1480000- 1930000 2^57
1930000- 2360000 2^58
2360000- 2950000 2^59
2950000- 3960000 2^60
3960000- 5160000 2^61
5160000- 6515000 2^62
6515000- 8250000 2^63
8250000-13380000 2^64
13380000-17890000 2^65
17890000-21590000 2^66
21590000-28130000 2^67
28130000-35200000 2^68
35200000-44150000 2^69
44150000-57020000 2^70
57020000-71000000 2^71
71000000-79300000 2^72
> What must i do if i want to continue testing my exponent above
> 2^67.
Any further trial factoring above 2^67 will be less efficient, from
an overall GIMPS throughput perspective, in determining whether that
Mersenne number is prime than simply switching to the L-L test.
You'll help GIMPS's progress most by sticking to the
program-assigned factoring limit.
Also, you can't continue trial factoring above 2^67 on your exponent
unless you give up using PrimeNet for assignments.
If you still really, really want to do it, then read all the
documentation files that came with your Prime95 software. The
answer's in there somewhere, sort of. I won't spell it out here.
Richard Woods
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Date: Thu, 18 Mar 2004 18:40:06 +0100
From: [EMAIL PROTECTED]
Subject: Mersenne: Incoming message
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End of Mersenne Digest V1 #1106
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