Amicable Numbers is an independent research project that uses Internet-connected computers to find new amicable pairs. You can contribute to our research by running a free program on your computer.

Current goal of the project is to find all amicable pairs with smallest member < 10²¹.
All new findings are published regularly on the Amicable pairs list page.

Though mathematics is not an experimental science, mathematicians often look for examples to test conjectures (which they then hope to prove). As the number of examples increases, so does (in a sense) their understanding of the distribution.

The goal is to collect all amicable numbers up to a very large limit. This data will facilitate theoretical research in the field: 1, 2, 3 to name a few examples. It will also help to improve understanding of the properties of Divisor function.

Join Amicable Numbers

MarkRBright   
Just another aging techie enjoying a bit of Boincing.

The search up to 10^21 is finished
After more than 6 years, the BOINC project to find all amicable numbers up to 10²¹ is finished!

There are 4,810,863 21-digit amicable numbers, and 4,661,814 (96.9%) of them were found by BOINC volunteers. More than 60 million WUs were created and crunched through over the years.

Congratulations everyone and thank you all for your participation!

There are no plans to run the search up to 10²² at the moment - it will take more than 60 years, which is impractical. If, and that's a big IF, a new more efficient exhaustive search method is discovered in the future, then it will make sense to start the project again.
20 Apr 2026, 8:17:02 UTC · Discuss

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Unplanned maintenance
Workunit and task distribution is temporarily off until the backlog of WUs clears.

Edit: Maintenance finished, everything should be back to normal now.
4 Oct 2025, 10:57:49 UTC · Discuss

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WU size and credits are doubled now
First of all, thanks to everyone participating in the project!

I've noticed today that solved work units come in so fast that the server can't keep up. I had to double the size of each work unit (together with the credit for it) to reduce the load on the server.

Keep crunching! Thanks again!
28 Oct 2024, 13:25:04 UTC · Discuss

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Maintenance on 2024-11-07 (7th of November)
Project server will be unavailable on 2024-11-07 (7th of November), from 03:30 AM UTC to 05:30 AM UTC due to the maintenance in a Hetzner datacenter where the server is hosted.
24 Oct 2024, 6:52:36 UTC · Discuss

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Part 3 of the 10^21 search has started
Part 2 of the 10²¹ search will be finished, for the most part, today. There are still ~2000 unfinished WUs which can take up to a couple of weeks to complete, depending on individual participants processing them.

Part 3 of the search will look for amicable pairs where the smaller member of the pair has its largest prime factor between 10¹¹ and 10¹⁴:

m=m1*p, 10¹¹ < p < 10¹⁴

There are too many primes larger than 10¹⁴ (all primes up to 5*10¹⁹ need to be checked), so they will require a different approach and a new application when the time comes.

Most of the participants have already started getting part 3 WUs. The new work units will require ~2.5-2.7 GB RAM on CPU, and the same amount of RAM on GPU (for GPU applications), so the minimum requirement for GPUs will be 3 GB RAM.

Update April 9th, 2024: I found a bug in OpenCL applications, which resulted in many false negatives (around 12.5%). The bug was in the new code that I added for the part 3, so the previous search is unaffected. Version 3.09 has this bug fixed, but I had to restart the part 3. Luckily, we were only a couple of weeks in.
27 Mar 2024, 10:24:42 UTC · Discuss

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Beta test of part 3 of the 10^21 search
Part 2 of the 10^21 search will be over some time in March this year, so it's time to start getting ready for part 3! I found a few bugs in the current GPU applications that needed to be fixed (CPU applications are fine).

To make sure everything works as intended, I will launch a small beta-test of the part 3 search by the end of this week. The new work units will require ~2.5-2.7 GB RAM on CPU, and the same amount of RAM on GPU (for GPU applications), so the minimum requirement for GPUs will be 3 GB RAM.

The purpose of the test is to confirm that all OS/CPU/GPU combinations work properly and confirm that 3 GB GPUs can process the new work units.
22 Jan 2024, 20:15:18 UTC · Discuss

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The search up to 10^21
The search up to 10²¹ has started! Over 50,000 new amicable pairs have been found in the first few days.

Due to the huge size of search space, first stage of the search will only look for all amicable pairs where smaller number is of the form 3^N*...*p where N > 0 and p < 10¹¹. I expect that this will find 2-3 million new amicable pairs in a year or so.

Note that system memory requirements are higher now because all prime numbers < 10¹¹ need to be stored in memory for the search.
25 Oct 2019, 13:11:37 UTC · Discuss

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GPU version bugfix release
GPU versions for all platforms were updated to fix "Error -58" computation error when resuming stopped task.
8 Apr 2019, 11:00:47 UTC · Discuss

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Power outage
The server was down today between 11:04 and 14:20 (CEST) because of a major power outage in Hetzner datacenter. Everything should be up and running again now.
24 May 2018, 12:41:34 UTC · Discuss

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GPU version fix
There was a serious bug in GPU version: https://github.com/SChernykh/Amicable/commit/3470ed855d244100f37b0b7ab4c912d2b9365a93
Unfortunately, it skipped some numbers in an unpredictable way for "large primes" work units for the last 20 days 9 days. I'll have to reissue all such work units to cover everything missed. The GPU versions for all platforms are already updated.

Sorry for the mess, but it's good that the bug is fixed relatively early, because it was hard to notice.
27 Mar 2018, 17:45:25 UTC · Discuss

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