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 2015-01-11, 19:54 #45 EdH     "Ed Hall" Dec 2009 Adirondack Mtns 3,253 Posts 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...
 2015-01-11, 21:05 #46 Drdmitry     Nov 2011 2·32·13 Posts 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!
2015-01-12, 21:09   #47
Drdmitry

Nov 2011

2·32·13 Posts

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

2015-01-12, 21:58   #48
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

29·313 Posts

Quote:
 Originally Posted by Drdmitry 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!

2015-01-13, 00:32   #49
EdH

"Ed Hall"
Dec 2009

3,253 Posts

Quote:
 Originally Posted by Drdmitry Quite unexpectedly a program have found a new cycle:
Excellent!! Congrats!

2015-03-08, 10:56   #50
AndrewWalker

Mar 2015
Australia

10100102 Posts

Quote:
 Originally Posted by Drdmitry 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

2015-03-09, 09:56   #51
Drdmitry

Nov 2011

2·32·13 Posts

Quote:
 Originally Posted by AndrewWalker 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
 newfound_druz.txt (30.9 KB, 78 views)

Last fiddled with by Drdmitry on 2015-03-09 at 09:56

 2015-03-10, 22:50 #52 Maciej   Mar 2015 710 Posts 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??
 2015-03-11, 09:10 #53 AndrewWalker     Mar 2015 Australia 2×41 Posts 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
2015-03-11, 09:58   #54
Drdmitry

Nov 2011

2×32×13 Posts

Quote:
 Originally Posted by AndrewWalker 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?

 2015-03-11, 11:15 #55 AndrewWalker     Mar 2015 Australia 2×41 Posts 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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