What is Amicable Numbers?

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.

What is the end goal of Amicable Numbers?
Why participate in it?

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.

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I've been a member of SETI.USA since 2009. We are an International Team of like minded individuals that wish to help the Scientific Community. Come and...

News

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

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

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 3N*...*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

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

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

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

MacOS GPU version released
GPU requirements are the same as for Windows and Linux versions: AMD (HD 5xxx or newer) or NVIDIA (GTX 4xx or newer) GPU with OpenCL support and at least 2 GB of video memory.

The MacOS GPU version has been confirmed to work on OS X El Capitan and MacOS Sierra. Older MacOS versions have poor OpenCL support and may fail to run the GPU version.
16 Sep 2017, 8:24:36 UTC · Discuss

The search up to 2^64 is complete!
Congratulations to everyone who took part in this search!

- There are 2,390,655 amicable pairs with smaller member below 2^64 in total
- BOINC volunteers found 552,874 new amicable pairs below 2^64

A more detailed analysis of amicable pairs distribution will follow later this month. And don't forget to edit your preferences and check that you're all set up to run the current search up to 10^20.
16 Aug 2017, 7:26:12 UTC · Discuss

Credits for the old application (search up to 2^64) doubled
It's sad to see everyone leave the old (and unfinished) application "Amicable Numbers up to 2^64" so I decided to double credit points for it.

Everyone can switch their CPUs back to gain some more credits.

Note that you can run "Amicable Numbers up to 2^64" on CPU and "Amicable Numbers up to 10^20" on GPU on the same PC at the same time!

Here's what you need to set in project preferences:
Use CPU, AMD/ATI GPU, NVIDIA GPU = yes
Run application 2^64 = yes
Run application 10^20 = no
Accept work from other applications = yes

With these settings your CPU will get tasks (and more credits in general) from "2^64" only, and your GPU will get tasks from "10^20" thanks to "Accept work from other applications = yes" selected.
3 Jul 2017, 16:09:14 UTC · Discuss

The search up to 10^20
It will be started tomorrow (July 1st)!

I've decided to speed up the launch of the next search due to performance issues with GPU version that everyone experiences now.

What will happen on July 1st:

1) "Amicable Numbers up to 10^20" application will start receiving tasks.
2) All GPU versions for the old application "Amicable Numbers up to 2^64" will be deprecated.
3) Credit points for both applications will be re-balanced to make sure that RAC doesn't change.

So all GPUs should automatically switch to the new application, and the old application will remain as CPU-only until it finishes.
30 Jun 2017, 16:11:33 UTC · Discuss

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