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Old 2006-11-24, 15:35   #1
grandpascorpion
 
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Default Multiply Perfect Numbers

Hello,

I'm curious about the methods for optimizing the search for these. Could anyone lend some insight? I see George Woltman has found a large number of these.
Thanks.

Some background: http://wwwhomes.uni-bielefeld.de/achim/mpn.html
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Old 2006-11-24, 16:34   #2
Prime95
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There used to be a mailing list but it died off.

At present I am the only active searcher. I have a dual PII-400 that cannot run prime95 due to sumout errors searching 24/7.

In simplest terms, the program basically does the following:

1) Generate a likely combination of powers for the first 5 primes. (actually the number of primes is configurable). Add up all the sigma chains for these 5 primes.

2) Now we play whack-a-mole with the next 34 primes. For example, if the sigma chains for the first 5 primes generated 47^2 then we add in the sigma chain for 47^2, then we pick another random prime and add in its sigma chain, say 59^3, this sigma chain might generate another 47, so we'll have to go back and change 47^2 to 47^3 at some point. When there are no more powers to correct among these 34 primes, then we proceed to step 3 (or abort this step after trying several thousand correction attempts).

3) We now check if the perfectly balanced sigma chains of the 34 primes happens to magically match the powers of the first 5 primes. If so, we have a multi-perfect number.

You can see that this is not a foolproof search. Many multiperfects are missed or found only after many attempts.

I have ugly source code that you can use to start your own research. I can even give you some ideas to try (or better yet you can invent some new ones)!
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Old 2006-11-24, 16:53   #3
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Thanks much for the explanation, George. I'd appreciate if you could point me to the source code (and share your ideas as well). Is it all C?

Last fiddled with by grandpascorpion on 2006-11-24 at 17:01
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Old 2006-11-24, 17:51   #4
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Here is a good resource:


http://adt.lib.uts.edu.au/public/adt...311/index.html



Joseph
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Old 2006-11-24, 19:38   #5
Prime95
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I'm uploading mpfn.zip to ftp://mersenne.org/gimps

It is C++ code. The file mpfn.cpp is the one to look at. It reads chains.db and subst.db which are produced by the massage*.cpp files.

Ideas:

1) Ditch all the nasty tree pruning code for the small primes. This was a classic mistake where I optimized the code before taking the time to determine if it was the bottleneck. It isn't.

2) Use SSE or SSE2. The small prime powers could be kept in an array of bytes with sigma chain additions and subtractions done brutally fast using these instructions.

3) Try different heuristics in the whack-a-mole code for deciding if we are getting closer to a solution.

4) Sometimes, I think, some power combinations are hard to generate using the whack-a-mole technique. For example, lets say you're at 47^3 and 101^2. Yet, 101^3 generates another 47 and 47^4 generates another 101. The whack-a-mole code will never explore this case because 47 and 101 are currently in balance.

5) Maybe it would make more sense to have more than 34 mid-size primes. It would be nice if you could change the number of small primes using a runtime parameter (rather than recompiling and rebuilding the chains.db).

It has been several years since I last looked at the code, so I'm sure there are more possible ideas that I've long since forgotten.

Last fiddled with by Prime95 on 2006-11-24 at 20:01
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Old 2006-11-24, 19:47   #6
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Thanks guys. I'll check both out.
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