Amicable pairs news

October 16th, 2023

New record amicable pair with the smallest known m/n ratio discovered:

X84 "BOINC: Science United, BudgieSoft" 2023
149251425686668100950=2*5^2*7^2*11*17*23*31*313*523*1487*1877
255075095917125253802=2*31*53*313*743*4463*74789471

The ratio is ~0.58512739, previous smallest known m/n ratio was ~0.59834304 for the odd-odd pair (92371445691525,154378742017851) discovered in 1997 and ~0.60512272 for the even-even pair (523445399739689350,865023542517044858) discovered in 2015

December 27th, 2022

New record for the largest known amicable pair: a 56259-digit pair was added today!

November 1st, 2019

The BOINC project to find all amicable numbers up to 10²⁰ is complete! A new BOINC search to find all amicable numbers up to 10²¹ has started: https://sech.me/boinc/Amicable/

August 16th, 2017

The BOINC project to find all amicable numbers up to 2⁶⁴ is complete! A new BOINC search to find all amicable numbers up to 10²⁰ has started: https://sech.me/boinc/Amicable/

June 28th, 2017

New record for the largest known amicable pair: a 56250-digit pair was added today. This pair was found by Paul Jobling.

January 30th, 2017

A distributed BOINC project to find all amicable pairs with smaller member below 2⁶⁴ has started. Everyone can join it: https://sech.me/boinc/Amicable/

August 16th, 2016

All amicable pairs with smaller member below 10¹⁸ are now known! There are 901,312 such pairs, 485,789 of them are 18-digit. It took 3.5 months (20 PCs running 24/7) to extend the border from 10¹⁷.

May 10th, 2016

New article added: A collection of methods for finding type (i, 1) amicable pairs/breeders

April 11th, 2016

The 1,000,000,000th amicable pair has been submitted today!!!

April 4th, 2016

Submission form now uses compression to speed up huge submissions.

March 7th, 2016

The first type (8, 1) amicable pair has been found! Both pair members are 80 digits long:

81 Chernykh 2016
14438796350376953678174322791271636288778421212339464839728409858893107308551670=2*5*13*59*71*73*1061*3019*4211*65521001603*4841941814273029742117557*84877110421196263915758493
14865176913743789894174101450879657763077918214583046351774081022511427255902730=2*5*13*59*1938093469849255527271721180036461246815895464743552327480323470992363397119

Type (i, 1) amicable pairs have a long history:

first (2, 1) pair was known to Pythagoras in ~500 B.C
first (3, 1) pair was discovered by Poulet in 1941
first (4, 1) pair was discovered by Lee in 1966
first (5, 1) pair was discovered by Te Riele in 1982
first (6, 1) pair was discovered by Pedersen in 1997
first (7, 1) pair was discovered by Garcia in 2001

Each step is exponentially harder to take, so it took another 15 years to find the first (8, 1) amicable pair.

January 30th, 2016

The first examples of amicable pairs where the two members have different smallest prime factors have been found!
First such pair was found on October 8th, 2015 but went unnoticed until now. Today there are 7 such pairs known:

X43 Chernykh 08 Oct 2015 22:54:20 GMT
445953248528881275=3^2*5^2*7*13*19*37*43*73*439*22483
659008669204392325=5^2*7*13*19*37*73*571*1693*5839

X43 Chernykh 02 Nov 2015 10:54:58 GMT
1097581690986390225=3^3*5^2*7^3*11^2*31*61*4507*4597
1615281291559017775=5^2*7^3*11^2*31*43*2851*409639

54 Chernykh 26 Oct 2015 17:32:43 GMT
1281431098689616875=3*5^4*7*11^2*13*19*373*2503*3499
1769164614201263125=5^4*7*11^2*19*31*97*1699*34429

X43 Chernykh 08 Jan 2016 10:14:39 GMT
1993991197249826775=3^2*5^2*7*11^2*13*109*307*1567*15349
2901232579265245225=5^2*7*11^2*13*307*439*4297*18199

X43 Chernykh 29 Nov 2015 07:59:53 GMT
4438032132485808675=3^2*5^2*7^3*13*31*43*61*1291*42139
6288132595256655325=5^2*7^3*13*31*37*43*103*11103889

X54 Chernykh 31 Oct 2015 09:54:31 GMT
8265325617300085425=3^3*5^2*7*13*19*31*151*727*937*2221
12234092897366205775=5^2*7*13*19*31*109*1567*1741*30703

X54 Chernykh 09 Jan 2016 01:57:14 GMT
9759636606249354825=3^2*5^2*7*11^2*13*139*241*6967*16879
14162430025984014775=5^2*7*11^2*13*223*439*7369*71317

January 20th, 2016

Fixed downloading of very large files. Files larger than 500 MB are now split into parts.

January 19th, 2016

The site moved to a new, much better server. Submission form is operational again.

December 20th, 2015

New article added: A million new amicable pairs every day: some new methods for finding amicable pair breeders

December 8th, 2015

Amicable pair submission form is updated:

It properly handles "time limit exceeded" cases now and automatically resumes from the point where it timed out
It's also a tiny bit faster now

November 11th, 2015

New article added: Amicable pairs: searching for them efficiently

October 30th, 2015

The first amicable pair of type (10,x) found:

10,5 Chernykh 2015
5963619754366409564=2^2*11*13*19*47*67*79*149*179*191*433
7853966263527990436=2^2*251*1151*2549*15359*173599

All other currently known amicable pairs have type (i,j) where both i and j are less than 10.

October 21st, 2015

New record for the largest known amicable pair: a 43145-digit pair was added today.
This pair was originally found by Andrew Walker and Paul Jobling on September 26th, 2011.

October 14th, 2015

New record even-even amicable pair with the smallest known m/n ratio found:

X73 Chernykh 2015
523445399739689350=2*5^2*7*11*17*31*37*73*191*500083
865023542517044858=2*31*37*107*63487*55509323

The ratio is ~0.60512272, previous record ratio was ~0.6468 for the pair (35049418250,54192685558)

October 12th, 2015

Submission form was updated: it will now provide a download link to a list of all new amicable pairs upon submission.

October 4th, 2015

I've added all pairs from Jan Munch Pedersen's site and Patrick Costello's site to this amicable pairs database.
The site is up and running now!

Sergei Chernykh