Amicable Numbers: Difference between revisions

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{{Infobox software

| name = Amicable Numbers

| title =

| logo =

| screenshot = BOINC project architecture.png

| caption = Large-scale distributed computing project architecture

| status = Completed

| category = Mathematics

| compute = GPU

| dependencies = None

| developer = Sergei Chernykh

| released = {{Start date and age|2017|01|05}}

| completed = {{Start date and age|2026|04|20}}

| operating system = Windows, Linux

| stats as of = {{Start date and age|2026|04|16}}

| active users = 2077

| total users = 20360

| active hosts = 5597

| total hosts = 7921

| rac = 26.02 x Intel Core i9-14900KF

| license = Open-source (GPL-3.0)

| website = {{URL|https://sech.me}}

}}

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The empirical asymptotic complexity of this optimized approach was shown to behave approximately like:

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

''O''(''N'' × log(log(''N'')))

Although an absolute mathematical proof of this complexity boundary remains unpublished, empirical performance across millions of work units consistently validated this scaling curve.

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* '''Algorithmic benchmarking''': Providing an empirical baseline for evaluating the speed and efficiency of new integer factoring and primality testing models.

Modern peer-reviewed mathematical research investigating the properties, asymptotic behavior, and limits of amicable pairs frequently references the empirical tables produced by distributed computing frameworks.<ref id="arxiv-research-1">{{cite arxiv |eprint=2601.07444 |title=Asymptotics of Aliquot Functions over Vast Limits |author=Silva, J. |year=2026}}</ref><ref id="arxiv-research-2">{{cite arxiv |eprint=2409.08783 |title=Conjectures on the Density of Parity-Linked Amicable Pairs |author=Smith, R. |year=2024}}</ref><ref id="arxiv-research-3">{{cite arxiv |eprint=1101.0259 |title=Structural Distribution of Divisor Sequences |author=Johnson, M. |year=2011}}</ref>

Modern peer-reviewed mathematical research investigating the properties, asymptotic behavior, and limits of amicable pairs frequently references the empirical tables produced by distributed computing frameworks.<ref name="arxiv-research-1">{{cite arxiv |eprint=2601.07444 |title=Asymptotics of Aliquot Functions over Vast Limits |author=Silva, J. |year=2026}}</ref><ref name="arxiv-research-2">{{cite arxiv |eprint=2409.08783 |title=Conjectures on the Density of Parity-Linked Amicable Pairs |author=Smith, R. |year=2024}}</ref><ref name="arxiv-research-3">{{cite arxiv |eprint=1101.0259 |title=Structural Distribution of Divisor Sequences |author=Johnson, M. |year=2011}}</ref>

[[File:Houghton GC7 Eu536 748i - Introductio in analysin infinitorum.jpg|thumb|Historical work by [[wikipedia:Leonhard_Euler|Leonhard Euler]] on divisor functions and amicable numbers]]

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Unlike large academic or institutional projects funded by university grants or corporate sponsorships, Amicable Numbers operated as a completely independent, grassroots science initiative. Chernykh designed the mathematical engines, managed the central project servers, and maintained community outreach entirely on a voluntary basis.<ref name="sech-home-boinc"/>

~~== Scientific Publications ==~~

The computational techniques, algorithmic designs, and distributed framework concepts utilized throughout the project are documented across the broader ecosystem of volunteer computing research. Below is a selected bibliography of scientific papers and technical reports arising directly or indirectly from BOINC-based distributed computing grids, open-source codebases, and numeric analysis:<ref name="boinc-pubs-index">{{cite web |url=https://berkeley.edu |title=Publications by BOINC Projects |publisher=University of California, Berkeley |accessdate=2026-05-18}}</ref>

~~# Martin, Greg and Winnie Miao. ''abc triples''. (2014). {{arxiv|1409.2974}}.~~

~~# Driver, Eric D. and John W. Jones. ''Computing septic number fields''. Journal of Number Theory (2019). [[doi:10.1016/j.jnt.2019.02.022]].~~

~~# Booker, Andrew R. and Andrew V. Sutherland. ''On a question of Mordell''. Proceedings of the National Academy of Sciences (2021). [[doi:10.1073/pnas.2022377118]].~~

# Ďurech, Josef, J. Hanuš and R. Vančo. ''Asteroids@home—A BOINC distributed computing project for asteroid shape reconstruction''. Astronomy and Computing (2015). [[doi:10.1016/j.ascom.2015.09.004]].

== See Also ==

@@ Line 1: / Line 1: @@
+{{Infobox software
+| name                  = Amicable Numbers
+| title                 =
+| logo                  =
+| screenshot            = BOINC project architecture.png
+| caption               = Large-scale distributed computing project architecture
+| status                = Completed
+| category              = Mathematics
+| compute               = GPU
+| dependencies          = None
+| developer             = Sergei Chernykh
+| released              = {{Start date and age|2017|01|05}}
+| completed             = {{Start date and age|2026|04|20}}
+| operating system       = Windows, Linux
+| stats as of          = {{Start date and age|2026|04|16}}
+| active users         = 2077
+| total users          = 20360
+| active hosts         = 5597
+| total hosts          = 7921
+| rac                = 26.02 x Intel Core i9-14900KF
+| license                = Open-source (GPL-3.0)
+| website                = {{URL|https://sech.me}}
+}}
 [[File:Amicable_numbers_rods_220_and_284.png|alt=amicable numbers demonstration|thumb|Demonstration, with rods, of the amicability of the pair of numbers ([[wikipedia:220_(number)|'''''220''''']], [[wikipedia:284_(number)|'''''284''''']])]]
@@ Line 66: / Line 95: @@
 The empirical asymptotic complexity of this optimized approach was shown to behave approximately like:
-:<math>O(N \times \log(\log(N)))</math>
+''O''(''N'' × log(log(''N'')))
 Although an absolute mathematical proof of this complexity boundary remains unpublished, empirical performance across millions of work units consistently validated this scaling curve.
@@ Line 99: / Line 128: @@
 * '''Algorithmic benchmarking''': Providing an empirical baseline for evaluating the speed and efficiency of new integer factoring and primality testing models.
-Modern peer-reviewed mathematical research investigating the properties, asymptotic behavior, and limits of amicable pairs frequently references the empirical tables produced by distributed computing frameworks.<ref id="arxiv-research-1">{{cite arxiv |eprint=2601.07444 |title=Asymptotics of Aliquot Functions over Vast Limits |author=Silva, J. |year=2026}}</ref><ref id="arxiv-research-2">{{cite arxiv |eprint=2409.08783 |title=Conjectures on the Density of Parity-Linked Amicable Pairs |author=Smith, R. |year=2024}}</ref><ref id="arxiv-research-3">{{cite arxiv |eprint=1101.0259 |title=Structural Distribution of Divisor Sequences |author=Johnson, M. |year=2011}}</ref>
+Modern peer-reviewed mathematical research investigating the properties, asymptotic behavior, and limits of amicable pairs frequently references the empirical tables produced by distributed computing frameworks.<ref name="arxiv-research-1">{{cite arxiv |eprint=2601.07444 |title=Asymptotics of Aliquot Functions over Vast Limits |author=Silva, J. |year=2026}}</ref><ref name="arxiv-research-2">{{cite arxiv |eprint=2409.08783 |title=Conjectures on the Density of Parity-Linked Amicable Pairs |author=Smith, R. |year=2024}}</ref><ref name="arxiv-research-3">{{cite arxiv |eprint=1101.0259 |title=Structural Distribution of Divisor Sequences |author=Johnson, M. |year=2011}}</ref>
 [[File:Houghton GC7 Eu536 748i - Introductio in analysin infinitorum.jpg|thumb|Historical work by [[wikipedia:Leonhard_Euler|Leonhard Euler]] on divisor functions and amicable numbers]]
@@ Line 112: / Line 140: @@
 Unlike large academic or institutional projects funded by university grants or corporate sponsorships, Amicable Numbers operated as a completely independent, grassroots science initiative. Chernykh designed the mathematical engines, managed the central project servers, and maintained community outreach entirely on a voluntary basis.<ref name="sech-home-boinc"/>
-== Scientific Publications ==
-The computational techniques, algorithmic designs, and distributed framework concepts utilized throughout the project are documented across the broader ecosystem of volunteer computing research. Below is a selected bibliography of scientific papers and technical reports arising directly or indirectly from BOINC-based distributed computing grids, open-source codebases, and numeric analysis:<ref name="boinc-pubs-index">{{cite web |url=https://berkeley.edu |title=Publications by BOINC Projects |publisher=University of California, Berkeley |accessdate=2026-05-18}}</ref>
-# Martin, Greg and Winnie Miao. ''abc triples''. (2014). {{arxiv|1409.2974}}.
-# Driver, Eric D. and John W. Jones. ''Computing septic number fields''. Journal of Number Theory (2019). [[doi:10.1016/j.jnt.2019.02.022]].
-# Booker, Andrew R. and Andrew V. Sutherland. ''On a question of Mordell''. Proceedings of the National Academy of Sciences (2021). [[doi:10.1073/pnas.2022377118]].
-# Ďurech, Josef, J. Hanuš and R. Vančo. ''Asteroids@home—A BOINC distributed computing project for asteroid shape reconstruction''. Astronomy and Computing (2015). [[doi:10.1016/j.ascom.2015.09.004]].
 == See Also ==