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Old 2005-05-29, 19:03   #1
jasong
 
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"Jason Goatcher"
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Default amicable, perfect, etc. numbers

I am wondering if anybody would like to write a program to help discover what I like to call "Numerology" numbers. Numbers which may or may not be useful today, but had religious significance(I think) many, many years ago.

I have noticed that a large majority of mathematics DC programs deal with prime numbers, with the possible exception of one or two. Would anyone be interested in developing a computer program that looks for special properties other than being prime?

I am sure anyone with a number theory book and the programming skills could whip something together within the week.

Would anyone like to try?
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Old 2005-05-30, 07:54   #2
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I wrote many programs about pairs and chains of amicable and perfect numbers some years ago...



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Old 2005-05-30, 19:50   #3
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Quote:
Originally Posted by ET_
I wrote many programs about pairs and chains of amicable and perfect numbers some years ago...



Luigi
So, you're saying it's already been done?
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Old 2005-05-30, 23:59   #4
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Of course, even perfect numbers are given by 2^(p-1)*(2^p-1) where 2^p-1 is a Mersenne prime, so GIMPS is just an organized search for even perfect numbers. As for odd perfect numbers, William (WBLIPP) is organizing a distributed search at www.oddperfect.org. He hopes to get that site set up with software to download this summer, but even now it is an interesting site to take a look at. A lot of different strategies have been used to search for amicable numbers and thousands of amicable pairs are known, so you might want to do a little research before starting your own search.
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Old 2005-06-05, 17:40   #5
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Cool amicable, perfect, etc. numbers

Quote:
Originally Posted by ET_
I wrote many programs about pairs and chains of amicable and perfect numbers some years ago...



Luigi

It woud be a good idea for the newer lot that you revive these threads if it will benefit the other not so knowledgeable set and if its not too technical
Thanks Luigi.
Mally
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Old 2005-06-06, 20:28   #6
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Quote:
Originally Posted by mfgoode

It woud be a good idea for the newer lot that you revive these threads if it will benefit the other not so knowledgeable set and if its not too technical
Thanks Luigi.
Mally
It has been about 20 years ago... My first program to look for Armstrong numbers was written in Basic and run on a Commodore 64 It took a whole day to check the first 1,000 nubers.

Then I translated it into PC Basic and C, but I never opened a thread on Mersenneforum. Maybe at that time there was not even Internet in Italy...

The main problem was that numbers tended to grow very rapidly: I remember that my first C program didn't ever stop, because I used an unsigned integer of 16 bits and forgot to check for overflow. "Shat hippens"...

There are many different heuristics to speed up the search: you can revrite the factoring and powering algorithms instead of using the standard library, and try to work on moduli to avoid useless work...

It's been fun!

Luigi

Last fiddled with by ET_ on 2005-06-06 at 20:28
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Old 2005-06-08, 17:27   #7
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Cool amicable, perfect, etc. numbers

Thank you Luigi. Lets hope someone else takes up the topic and takes the
trouble to put it down.
Oh to be in Rome! My song? 'amore scusame!' very old , very romantic
Mally
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Old 2005-06-08, 17:58   #8
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Quote:
Originally Posted by mfgoode
Thank you Luigi. Lets hope someone else takes up the topic and takes the
trouble to put it down.
Oh to be in Rome! My song? 'amore scusame!' very old , very romantic
Mally
http://www.canzoni-mp3.net/amore_scusami.htm
http://www.lyrical.nl/song/25315



Luigi
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Old 2005-06-09, 15:16   #9
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Quote:
Originally Posted by ET_
It has been about 20 years ago... My first program to look for Armstrong numbers was written in Basic and run on a Commodore 64 It took a whole day to check the first 1,000 nubers.

Then I translated it into PC Basic and C, but I never opened a thread on Mersenneforum. Maybe at that time there was not even Internet in Italy...

The main problem was that numbers tended to grow very rapidly: I remember that my first C program didn't ever stop, because I used an unsigned integer of 16 bits and forgot to check for overflow. "Shat hippens"...

There are many different heuristics to speed up the search: you can revrite the factoring and powering algorithms instead of using the standard library, and try to work on moduli to avoid useless work...

It's been fun!

Luigi

Allow me to point out that substantial computation has already been done.
Check section B4 of Richard Guy's "Unsolved Problems in Number Theory".
He gives a summary of what is known as well as extensive references.

Anyone who wants to work in computational number theory should have
a copy of this book. I would also recommend owing a copy of Knuth Vol 2.,
H. Cohens book "A Course in Computational Algebraic Number Theory",
Crandall & Pomerance's "Prime Numbers: A computational Perspective",
and either Reisel's or Bressoud's books on Factoring .
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Old 2005-06-09, 15:50   #10
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I have the 2nd edition (1994) of Richard Guy's book, although I think that the 3rd edition has recently been issued. He says in the 2nd edition that all amicable pairs with the smaller member less than 2*10^11 have been computed by Moews & Moews, but doesn't give a reference. If you are interested in raising this bound, the paper by te Riele, "Computation of all the amicable pairs below 10^10", Math. Comput., 47(1986) 361-368 and S9-S40, might be a good place to start your research. One open question is whether or not there are any amicable pairs where one number is even and the other is odd.
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Old 2005-06-09, 16:49   #11
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Here are references to the Moews, D. and Moews, P. C. papers:

"A search for aliquot cycles below 10^10", Math. Comp. 57, 849-855, 1991

"A search for aliquot cycles and amicable pairs", Math. Comp. 61, 935-938, 1993
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