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Old 2015-01-11, 19:54   #45
EdH
 
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Latest Find:
Code:
28405878617032 = 2E3.7.17.317.94126523
36247748188088 = 2E3.23.131.1503806347
35213129468872 = 2E3.53.191.2417.179899
32437790595128 = 2E3.3391.14747.81083
@Drdmitry:

I was unaware there may be undiscovered amicable pairs within the regions we're examining. I also am not sure if I still have all of the ones I've found, since I've been concentrating on 4 cycle and (although there shouldn't be any), I occasionally check to see if any 3 cycles turned up. The only amicable work I had been doing was using the known lists on occasion to validate changes in my code, but since I've started finding 4 cycles, I now use those. Maybe I'll toss something together to do a check on what I may have against the lists...
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Old 2015-01-11, 21:05   #46
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Ed, as far as I know, the exhaustive search of amicable pairs has been made up to 10^14. So in the range you are working with there should not be any new undiscovered amicable pairs.

By the way, congratulations with all your last findings!
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Old 2015-01-12, 21:09   #47
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Quite unexpectedly a program have found a new cycle:
Quote:
166372740802275 = 3*5^2*7*19*31*538031339
175212596336925 = 3*5^2*7*31*599*3617*4969
167269483407075 = 3*5^2*7^2*19*31^2*1279*1949
183095234672925 = 3*5^2*19*29*31*37*3862787
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Old 2015-01-12, 21:58   #48
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Quote:
Originally Posted by Drdmitry View Post
Code:
166372740802275 = 3*52*7*19*31*538031339
175212596336925 = 3*52*7*31*599*3617*4969
167269483407075 = 3*52*72*19*312*1279*1949
183095234672925 = 3*52*19*29*31*37*3862787
This is very odd.
Congrats!
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Old 2015-01-13, 00:32   #49
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Quote:
Originally Posted by Drdmitry View Post
Quite unexpectedly a program have found a new cycle:
Excellent!! Congrats!
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Old 2015-03-08, 10:56   #50
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Quote:
Originally Posted by Drdmitry View Post
Unfortunately I do not have enough computer power to do a exhaustive search of aliquot cycles in a reasonable range. So I played a little bit with my program to see what can be done with it relatively quickly.
Firstly if the search is restricted to odd numbers only then it goes much faster: It took about 12.3 secs per 10^8 on one core of my laptop - about 100 times faster than for even numbers. I checked all odd numbers between 10e13 and 12e13.
Then I also checked all 15-digit odd numbers of the form
3a1 * 5a2 * ... * 29a9 * n
with n up to around 6e7. I am still running a program to extend the full search for n up to about 12e8. However now as the new term starts I will not be able to pay too much attention on it and I am not sure that I will actually complete the search.
I did also some other funny checks. No new cycles of length more than two were found, I'm afraid. On the other hand 137 new amicable pairs, not presented in Pedersen's tables were found. I attach the file with them here.
I also found an inaccuracy in Mathworld article about amicable pairs. It is said that the smallest known pair with elements coprime to 6 has 32 digits. However in Pedersens list there is a pair

which has only 16 digits - twice less.
Hi there I only found this thread a few days back. That pair with 16 digits coprime to 6 I found way back in 2003? with David Einstein. It hasn't been proven as the smallest but I think there's a very good chance it is. We were doing a lot of searching in the 14-16 digit range. I hope to get back in touch with Jan somehow as a few years ago I sent him another record size pair and never got a reply.
Just this year I've started looking in the 15, 16 digit range again. To confirm what was mentioned above David finished the 14 digit amicables. From the numbers on Jans old page it looks like the 15d is probably over 90% done. I hope we can get you pairs and any other new ones incorporated into the lists!

Andrew
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Old 2015-03-09, 09:56   #51
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Quote:
Originally Posted by AndrewWalker View Post
Hi there I only found this thread a few days back. That pair with 16 digits coprime to 6 I found way back in 2003? with David Einstein. It hasn't been proven as the smallest but I think there's a very good chance it is. We were doing a lot of searching in the 14-16 digit range. I hope to get back in touch with Jan somehow as a few years ago I sent him another record size pair and never got a reply.
Just this year I've started looking in the 15, 16 digit range again. To confirm what was mentioned above David finished the 14 digit amicables. From the numbers on Jans old page it looks like the 15d is probably over 90% done. I hope we can get you pairs and any other new ones incorporated into the lists!

Andrew
Hi Andrew,

According to Pedersen's list that 16-digit number were found by Walker (You?) and Enstein in 2001. I can confirm that it is the smallest (and in fact the only 16 digits) amicable pair coprime with 6. As one of the tests I searched for all such pairs up to 10^16
At the moment a search for odd aliquot cycles runs almost autonomously on one of my laptops. I just check it twice a week. It reached n up to 27e8 (with n defined in my previous post). At the moment 289 non-listed pairs were found. I attach the file with all of them. No more longer cycles were discovered.

As far as I can see you did the search for all 15-digits pairs such that the largest prime divisor of the least term of the pair is at most 7 digits. At least all except one new discovered pairs have the biggest prime divisor of the smallest element bigger than 10 millions. And one exception is divisible by 3^11.

It would be nice to join the efforts. With only my poor computer resources it will take probably half a year to finish the search of all odd 15 digits cycles.

Dmitry
Attached Files
File Type: txt newfound_druz.txt (30.9 KB, 78 views)

Last fiddled with by Drdmitry on 2015-03-09 at 09:56
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Old 2015-03-10, 22:50   #52
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Welcome
I have new algorithm , to find all amicable numbers <10^15. I send my pairs to http://math.eku.edu/amicable-pairs-download. Dou you know people are looking for amicable numbers 10^15??
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Old 2015-03-11, 09:10   #53
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Hi Dmitry, that Walker is me. The search we did on the 15d was up to 10 million for the largest factor of the first number, it should be complete but if you find any missing let me know! Also did a little on the 16 digits but that was very far from complete.

I've just finished a run on 15d and 16d with factors up to 10000 for the first number. Again some are missed for the higher 16 digits because it is near the precision limit. I'm now starting a higher search for the even 15 digits. and will later do the same for the odds even if it's just a double check on yours. I've no idea what the highest possible factor for these is.

Andrew
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Old 2015-03-11, 09:58   #54
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Quote:
Originally Posted by AndrewWalker View Post
Hi Dmitry, that Walker is me. The search we did on the 15d was up to 10 million for the largest factor of the first number, it should be complete but if you find any missing let me know! Also did a little on the 16 digits but that was very far from complete.

I've just finished a run on 15d and 16d with factors up to 10000 for the first number. Again some are missed for the higher 16 digits because it is near the precision limit. I'm now starting a higher search for the even 15 digits. and will later do the same for the odds even if it's just a double check on yours. I've no idea what the highest possible factor for these is.

Andrew
I have found only one 15d pair with the highest factor of the first number less than 10 million. It is
Code:
166804820674965 = 3^11*5*11*41*211*1979
170487987977835 = 3^5*5*11*149*461*185711
Perhaps it was missed because of the high power of three.
I guess, you only search for amicable pairs, no longer Aliquot cycles?
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Old 2015-03-11, 11:15   #55
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Interesting, I've got that written down as the very last of the odd pairs found, so it may have been
cut off accidentally before being sent in.

From my latest run for 16d

5903755662561705=3^11.5.13^2.29.311.4373
6041108390596695=3^11.5.13^2.89.467.971

5480828320492525=5^2.7^2.11.13.17.19.23.31.103.1319
5786392931193875=5^3.7.11.13.19.31.37.43.61.809

were the final two found. I'm only searching for pairs.

I'll try and get in touch with a few people from a while back next week.

Andrew
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