20050529, 19:03  #1 
"Jason Goatcher"
Mar 2005
3507_{10} Posts 
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? 
20050530, 07:54  #2 
Banned
"Luigi"
Aug 2002
Team Italia
4,813 Posts 
I wrote many programs about pairs and chains of amicable and perfect numbers some years ago...
Luigi 
20050530, 19:50  #3  
"Jason Goatcher"
Mar 2005
6663_{8} Posts 
Quote:


20050530, 23:59  #4 
"Phil"
Sep 2002
Tracktown, U.S.A.
3×373 Posts 
Of course, even perfect numbers are given by 2^(p1)*(2^p1) where 2^p1 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.

20050605, 17:40  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
amicable, perfect, etc. numbers
Quote:
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 

20050606, 20:28  #6  
Banned
"Luigi"
Aug 2002
Team Italia
4,813 Posts 
Quote:
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 20050606 at 20:28 

20050608, 17:27  #7 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
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 
20050608, 17:58  #8  
Banned
"Luigi"
Aug 2002
Team Italia
4813_{10} Posts 
Quote:
http://www.lyrical.nl/song/25315 Luigi 

20050609, 15:16  #9  
Nov 2003
16444_{8} Posts 
Quote:
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 . 

20050609, 15:50  #10 
"Phil"
Sep 2002
Tracktown, U.S.A.
1119_{10} Posts 
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) 361368 and S9S40, 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.

20050609, 16:49  #11 
"Phil"
Sep 2002
Tracktown, U.S.A.
3×373 Posts 
Here are references to the Moews, D. and Moews, P. C. papers:
"A search for aliquot cycles below 10^10", Math. Comp. 57, 849855, 1991 "A search for aliquot cycles and amicable pairs", Math. Comp. 61, 935938, 1993 
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