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	'''ABC@home''' was a [[wikipedia:volunteer computing\|volunteer computing]] project running on the [[BOINC]] platform that searched for '''abc-triples''', sets of numbers related to the [[wikipedia:abc conjecture\|abc conjecture]], one of the best known open problems in [[wikipedia:number theory\|number theory]].<ref name="wikipedia">{{Cite web\|title=ABC@Home\|url=https://en.wikipedia.org/wiki/ABC@Home\|website=Wikipedia\|access-date=2026-07-18}}</ref> The project was organized by the Mathematical Institute of [[wikipedia:Leiden University\|Leiden University]] in the Netherlands, together with the Dutch Kennislink science communication institute, and ran from 2006 until it ceased operations in 2015.<ref name="palenstijn2023">{{Cite journal\|last=Alvarez-Salazar\|first=Elise\|last2=Barrios\|first2=Alexander J.\|last3=Henaku\|first3=Calvin\|last4=Soller\|first4=Summer\|title=On abc triples of the form (1, c − 1, c)\|journal=arXiv preprint\|year=2023\|arxiv=2301.01376}}</ref>		'''[https://web.archive.org/web/20130331200159/http://abcathome.com/ ABC@home]''' was a [[wikipedia:volunteer computing\|volunteer computing]] project running on the [[BOINC]] platform that searched for '''abc-triples''', sets of numbers related to the [[wikipedia:abc conjecture\|abc conjecture]], one of the best known open problems in [[wikipedia:number theory\|number theory]].<ref name="wikipedia">{{Cite web\|title=ABC@Home\|url=https://en.wikipedia.org/wiki/ABC@Home\|website=Wikipedia\|access-date=2026-07-18}}</ref> The project was organized by the Mathematical Institute of [[wikipedia:Leiden University\|Leiden University]] in the Netherlands, together with the Dutch Kennislink science communication institute, and ran from 2006 until it ceased operations in 2015.<ref name="palenstijn2023">{{Cite journal\|last=Alvarez-Salazar\|first=Elise\|last2=Barrios\|first2=Alexander J.\|last3=Henaku\|first3=Calvin\|last4=Soller\|first4=Summer\|title=On abc triples of the form (1, c − 1, c)\|journal=arXiv preprint\|year=2023\|arxiv=2301.01376}}</ref>

	== Background ==		== Background ==

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{{Infobox software
| name = ABC@home
| logo = Abclogo.jpg
| logo caption = ABC@home project logo
| screenshot = ABC@Home.png
| caption =
| description = ABC@home was a completed Mathematics BOINC project run by Leiden University that searched for abc-triples related to the abc conjecture in number theory.
| status = Completed
| category = Mathematics
| compute = CPU
| dependencies = None
| developer = Mathematical Institute, Leiden University
| author = Bart de Smit
| sponsor = Dutch Kennislink Science Institute
| maintainer = Bart de Smit
| released = {{Start date and age|2006|11|02}}
| completed = {{Start date and age|2015|11|30}}
| discontinued =
| repository =
| programming language =
| operating system = Cross-platform
| size =
| stats as of = {{Start date and age|2011|03|15}}
| average performance = ~10 teraflops
| active users = 7,300
| total users =
| active hosts =
| total hosts =
| rac =
| credit per day =
| gpu performance =
| cpu performance = ~10 teraflops
| website = {{URL|http://www.abcathome.com/}}
| license = Proprietary
}}

'''ABC@home''' was a [[wikipedia:volunteer computing|volunteer computing]] project running on the [[BOINC]] platform that searched for '''abc-triples''', sets of numbers related to the [[wikipedia:abc conjecture|abc conjecture]], one of the best known open problems in [[wikipedia:number theory|number theory]].<ref name="wikipedia">{{Cite web|title=ABC@Home|url=https://en.wikipedia.org/wiki/ABC@Home|website=Wikipedia|access-date=2026-07-18}}</ref> The project was organized by the Mathematical Institute of [[wikipedia:Leiden University|Leiden University]] in the Netherlands, together with the Dutch Kennislink science communication institute, and ran from 2006 until it ceased operations in 2015.<ref name="palenstijn2023">{{Cite journal|last=Alvarez-Salazar|first=Elise|last2=Barrios|first2=Alexander J.|last3=Henaku|first3=Calvin|last4=Soller|first4=Summer|title=On abc triples of the form (1, c − 1, c)|journal=arXiv preprint|year=2023|arxiv=2301.01376}}</ref>

== Background ==
The abc conjecture, proposed by [[wikipedia:Joseph Oesterlé|Joseph Oesterlé]] and [[wikipedia:David Masser|David Masser]] in 1985, concerns triples of positive, pairwise coprime integers <math>a</math>, <math>b</math>, and <math>c</math> satisfying

<math>a + b = c.</math>

For such a triple, the '''radical''' of the product <math>abc</math> is defined as the product of the distinct prime factors of <math>abc</math>:

<math>\operatorname{rad}(abc) = \prod_{p \mid abc} p.</math>

A triple is called an '''abc-triple''' if <math>\operatorname{rad}(abc) < c</math>. The '''quality''' of an abc-triple is defined as

<math>q(a,b,c) = \frac{\log c}{\log \operatorname{rad}(abc)}.</math>

The abc conjecture states that for every <math>\varepsilon > 0</math>, only finitely many abc-triples satisfy <math>q(a,b,c) > 1 + \varepsilon</math>.<ref name="alvarez2023-full">{{Cite journal|last=Alvarez-Salazar|first=Elise|last2=Barrios|first2=Alexander J.|last3=Henaku|first3=Calvin|last4=Soller|first4=Summer|title=On abc triples of the form (1, c − 1, c)|journal=arXiv preprint|year=2023|arxiv=2301.01376}}</ref> A proof of the conjecture would have far-reaching consequences for number theory, including an asymptotic version of [[wikipedia:Fermat's Last Theorem|Fermat's Last Theorem]] and [[wikipedia:Faltings's theorem|Faltings's theorem]].

Cataloguing abc-triples by brute-force search, and studying their quality distribution, is one of the main ways mathematicians have gathered empirical evidence about the conjecture. ABC@home was created to carry out such a search at scale using volunteered computing power.<ref name="boincwiki">{{Cite web|title=ABC@home|url=https://boinc.berkeley.edu/wiki/ABC@home|website=BOINC|publisher=University of California, Berkeley|access-date=2026-07-18}}</ref>

== History ==
[[File:Leiden University Seal.png|thumb|right|Seal of Leiden University, whose Mathematical Institute organized ABC@home]]
The roots of ABC@home lie in the Intercity Number Theory Seminar held in Leiden in September 2005, where researchers including [[wikipedia:Hendrik Lenstra|Hendrik Lenstra]] discussed algorithms for enumerating abc-triples.<ref name="icseminar">{{Cite web|title=Intercity Number Theory Seminar|url=http://pub.math.leidenuniv.nl/~brightmj/ic/20050909.html|website=Universiteit Leiden|access-date=2026-07-18}}</ref> In 2006, the Mathematical Institute launched a public science outreach initiative called "Reken mee met ABC" ("Count along with ABC"), which combined public education about the abc conjecture with a distributed computing search for triples. The project and its educational aims were described by Bart de Smit and Gillien Geuze in a 2007 article for ''Nieuw Archief voor Wiskunde''.<ref name="geuze2007">{{Cite journal|last=Geuze|first=Gillien|last2=de Smit|first2=Bart|title=Reken mee met ABC|journal=Nieuw Archief voor Wiskunde|series=5th series|volume=8|issue=1|year=2007|pages=26–30|url=http://www.nieuwarchief.nl/serie5/pdf/naw5-2007-08-1-026.pdf}}</ref>

The distributed computing side of the project was launched as ABC@home on the BOINC platform, allowing volunteers to donate spare CPU time toward searching for abc-triples below successively larger bounds.<ref name="wikipedia" /> By March 2011, the project had more than 7,300 active participants from 114 countries, with a combined [[wikipedia:BOINC Credit System|BOINC credit]] total exceeding 2.9 billion and an aggregate throughput of roughly 10 teraflops.<ref name="statsarchive">{{Cite web|title=Detailed user, host, team and country statistics with graphs for BOINC|url=https://web.archive.org/web/20101120150315/http://boincstats.com/stats/project_graph.php?pr=abc_main|website=BOINCstats|archive-url=https://web.archive.org/web/20101120150315/http://boincstats.com/stats/project_graph.php?pr=abc_main|archive-date=2010-11-20|access-date=2026-07-18}}</ref>

In 2011, the project reached a major milestone: it completed an exhaustive search finding all abc-triples <math>(a,b,c)</math> with <math>c < 10^{18}</math>, a total of 14,482,065 triples.<ref name="palenstijn2023" /> The project continued searching for larger triples with <math>10^{18} \le c < 2^{63}</math> from 2012 to 2015, though this later phase was not exhaustive. By the time it ceased operations in 2015, ABC@home had found a total of 23,827,716 abc-triples.<ref name="palenstijn2023" /><ref name="desmit">{{Cite web|last=de Smit|first=Bart|title=ABC Triples|url=https://www.math.leidenuniv.nl/~desmit/abc/index.php?set=1|website=Universiteit Leiden|access-date=2026-07-18}}</ref>

== Data and results ==
The full lists of triples found by ABC@home, along with derived lists of high-quality and unbeaten triples, were made available by Bart de Smit on a Leiden University web page.<ref name="desmit" /> The description and analysis of the search algorithm, along with observations about the resulting list of triples, form the final chapter of Willem Jan Palenstijn's 2014 PhD thesis, ''Radicals in Arithmetic''.<ref name="palenstijnthesis">{{Cite web|last=Palenstijn|first=Willem Jan|title=Radicals in Arithmetic|url=https://hdl.handle.net/1887/25833|website=Leiden University Scholarly Publications|type=PhD thesis|publisher=Universiteit Leiden|date=2014-05-22|access-date=2026-07-18}}</ref> The implementation of the BOINC client-side search application was separately addressed in a bachelor thesis by Thijs van Dijk in 2011.<ref name="desmit" />

The abc-triple with the highest known quality, <math>(2,\, 3^{10} \cdot 109,\, 23^5)</math>, has a quality of approximately 1.6299 and remains one of the most frequently cited examples in the literature on the conjecture.<ref name="alvarez2023-full" />

== Related projects ==
A separate, unrelated project called '''ABC Lattices@Home''' also searched for abc-triples using BOINC, but took a different approach: rather than an exhaustive search of a numerical range, it used a specialized lattice-based algorithm and educated guesses to look for good triples beyond <math>2^{100}</math>.<ref name="lattices">{{Cite web|title=ABC LATTICES@HOME|url=https://forum.boinc-australia.net/index.php?board=176.0|website=BOINC-Australia Forum|access-date=2026-07-18}}</ref>

== Publications ==

=== Papers based on ABC@home data ===
* {{Cite journal|last=Martin|first=Greg|last2=Miao|first2=Winnie|title=abc triples|journal=Functiones et Approximatio Commentarii Mathematici|volume=55|issue=2|year=2016|pages=145–176|arxiv=1409.2974}} A catalog of known abc-triples, both computational examples and theoretically constructed infinite families, together with a survey of the heuristics, refinements, and progress on the abc conjecture. [https://arxiv.org/abs/1409.2974 arXiv:1409.2974]
* {{Cite journal|last=Alvarez-Salazar|first=Elise|last2=Barrios|first2=Alexander J.|last3=Henaku|first3=Calvin|last4=Soller|first4=Summer|title=On abc triples of the form (1, c − 1, c)|journal=arXiv preprint|year=2023|arxiv=2301.01376}} Proves general results for constructing infinite families of abc-triples of the form <math>(1, c-1, c)</math>, and applies them to a direct statistical analysis of the 45,604 such triples with <math>c < 10^{18}</math> found by ABC@home.

=== Project documentation ===
* {{Cite web|last=Palenstijn|first=Willem Jan|title=Radicals in Arithmetic|url=https://hdl.handle.net/1887/25833|website=Leiden University Scholarly Publications|type=PhD thesis|publisher=Universiteit Leiden|date=2014-05-22|access-date=2026-07-18}} Chapter 5 describes the algorithm used by ABC@home to enumerate abc-triples and reports results from the project.
* {{Cite journal|last=Geuze|first=Gillien|last2=de Smit|first2=Bart|title=Reken mee met ABC|journal=Nieuw Archief voor Wiskunde|series=5th series|volume=8|issue=1|year=2007|pages=26–30|url=http://www.nieuwarchief.nl/serie5/pdf/naw5-2007-08-1-026.pdf}} Describes the founding and educational goals of the Reken mee met ABC project, of which ABC@home was the computational component.

== See also ==
* [[SRBase]]
* [[RakeSearch]]
* [[NumberFields@Home]]
* [[Ramanujan Machine]]

== References ==
{{Reflist}}

[[Category:Completed BOINC projects]]
[[Category:Mathematics BOINC projects]]

ABC@home - Revision history

Al Piskun: add archive link

Al Piskun: first light