Amicable Numbers

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Amicable Numbers was an independent research project that used BOINC based volunteer computing to search for new amicable pairs. The project was operated by mathematician and developer Sergei Chernykh and officially completed its large-scale search effort in April 2026 after more than six years of distributed computation.[1][2]

amicable numbers demonstration
Demonstration, with rods, of the amicability of the pair of numbers (220, 284)

Overview

Amicable Numbers focused on one of the oldest problems in number theory: discovering and cataloging amicable pairs. Two numbers are called amicable if the sum of the proper divisors of each number equals the other number.

The smallest and best-known amicable pair is:

  • 220 → 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
  • 284 → 1 + 2 + 4 + 71 + 142 = 220

The study of amicable numbers dates back to ancient Greece and the work of Pythagoras and later Arab mathematicians such as Thābit ibn Qurra. Leonhard Euler later discovered dozens of additional pairs in the 18th century.

Pythagoras, one of the earliest mathematicians associated with the study of amicable numbers

Why Amicable Numbers?

Collecting data on all amicable numbers up to a very large limit facilitates theoretical research in several fields:[3][4][5]

Data from this project also helped improve understanding of the properties of the divisor function, aliquot sums, and the statistical distribution of amicable pairs.

The project became one of the best-known BOINC mathematics projects due to its efficient algorithms and extremely large computational scale. It supported CPUs and GPUs on Windows, Linux, and macOS systems.[6]

Goal

The final goal of Amicable Numbers was to find all amicable pairs with smallest member less than 1021.

This enormous search space required years of distributed computing effort involving volunteers from around the world. More than 60 million BOINC work units were processed during the lifetime of the project.[7]

Completion of the Project

On April 20, 2026, project administrator Sergei Chernykh announced that the exhaustive search up to 1021 had been completed.[8]

The project reported:

  • 4,810,863 amicable numbers with 21 digits
  • 4,661,814 of them discovered through BOINC volunteers
  • Approximately 96.9% of all results contributed by volunteers
  • Over six years of continuous distributed computation

The administrator stated that there were no plans to continue searching up to 1022, because the estimated runtime with current algorithms would exceed 60 years.[9]

Following completion, the project stopped issuing new work units and transitioned into archival status.

Large-scale distributed computing projects such as Amicable Numbers rely on massive computational resources contributed by volunteers worldwide

Methods

BOINC was used because it was the only viable way for this independent project to obtain computational power at this scale.

Terminology

N - the limit for the search. The algorithm must find all amicable pairs with smaller member m ≤ N

m - smaller number in a pair

n - larger number in a pair, or an arbitrary natural number when used in enumerations (i.e. p1, ..., pn)

σ(m) - sum of all divisors of m, including 1 and m.

The following basic algorithm can be derived from the very definition of amicable pairs:

  • Iterate over all numbers between 1 and N
  • For each number m, calculate n=σ(m)-m
  • If n > m, then calculate σ(n)
  • If σ(n) = σ(m), then (m, n) is an amicable pair

This naive method is very inefficient because it requires one or two full factorizations per iteration. Several optimizations were therefore developed.[10]

Optimizations

  1. Avoiding factorization of m
  2. Trimming down factorization of n
  3. Reducing search space

As a result of these improvements, the project's algorithm — download the source code: Amicable.zip — was able to find all amicable pairs below 1013 in roughly 24 minutes on an Intel Core i7-4770K using eight parallel threads.

The asymptotic complexity appeared to behave approximately like:

<math>O(N \times \log(\log(N)))</math>

although no strict mathematical proof was published.

The Intel Core i7-4770K processor referenced in the project's performance benchmarks

GPU Computing

Amicable Numbers became notable in the BOINC community for supporting GPU acceleration in addition to traditional CPU computation.

The project supported:

  • NVIDIA GPUs
  • AMD GPUs
  • Intel GPUs
  • Multi-threaded CPU workloads

Community discussions often highlighted the project's unusually high memory usage and computational intensity compared to many other BOINC projects.[11]

Software and Source Code

The project's source code was publicly available and open for inspection.

GitHub repository:

The transparency of the software and algorithms helped build trust within the BOINC community and allowed independent verification of the project's methods.

Scientific Importance

Although amicable numbers are primarily an area of pure mathematics, exhaustive searches provide valuable empirical data for testing conjectures and studying the distribution of divisor-related functions.

Research related to amicable numbers includes:

  • divisor sums
  • aliquot sequences
  • abundant and deficient numbers
  • sociable numbers
  • computational number theory

Modern mathematical research on amicable numbers continues to appear in academic literature.[12][13][14]

Historical work by Leonhard Euler on divisor functions and amicable numbers

Community and BOINC Participation

The project attracted a dedicated international community of BOINC volunteers and competitive computing teams.

Several BOINC challenge events were organized around the project before its completion.[15]

Community discussions on Reddit and BOINC forums documented the project's final months and eventual completion.[16][17]

Project team / Sponsors

  • Sergei Chernykh

The project operated independently and was not affiliated with a university or corporate sponsor.

Scientific results

New data from the project was regularly published on the Amicable pairs list page during active operation.

By the completion of the project in 2026, the search up to 1021 represented one of the largest exhaustive amicable number searches ever performed using distributed volunteer computing.

See also

External links

References

  1. ↑ https://boinc.berkeley.edu/project_news.php
  2. ↑ https://sech.me/boinc/Amicable/
  3. ↑ https://math.dartmouth.edu/~carlp/Amicable1.pdf
  4. ↑ https://math.dartmouth.edu/~carlp/Amicable2.pdf
  5. ↑ https://math.dartmouth.edu/theses/undergrad/2014/Nguyen-thesis.pdf
  6. ↑ https://boinc.berkeley.edu/projects.php
  7. ↑ https://sech.me/boinc/Amicable/
  8. ↑ https://boinc.berkeley.edu/project_news.php
  9. ↑ https://sech.me/boinc/Amicable/
  10. ↑ https://sech.me/ap/articles.html#a1
  11. ↑ https://www.reddit.com/r/BOINC/comments/10nauyq
  12. ↑ https://arxiv.org/abs/2601.07444
  13. ↑ https://arxiv.org/abs/2409.08783
  14. ↑ https://arxiv.org/abs/1101.0259
  15. ↑ https://www.boincstats.net/
  16. ↑ https://www.reddit.com/r/BOINC/comments/1qfacih/end_of_an_era_amicable_numbers_boinc_project/
  17. ↑ https://www.reddit.com/r/BOINC/comments/1sis8a0/amicable_numbers_stuck_on_manager_list/
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