ABC@home

ABC@home
ABC@home project logo
Project
Status	Completed
Category	Mathematics
Compute	CPU
Requires	BOINC
Development
Developer	Mathematical Institute, Leiden University
Author	Bart de Smit, Willem Jan Palenstijn
Sponsor	Dutch Kennislink Science Institute
Maintainer	Bart de Smit
Initial release	November 2, 2006  (20 years ago)
Completed	November 30, 2015  (11 years ago)
Software
Operating system	Windows, Linux, Mac OS X (x86, x86-64, PowerPC)
BOINC statistics
Stats as of	8 November 2026
Performance	0 (project retired)
Active users	0
Total users	79,634
Active hosts	0
Total hosts	146,940
Analytics
RAC	0
Credit/day	0
GPU performance	None (CPU-only application)
CPU performance	0 (project retired)
Metadata
Website	http://www.abcathome.com/ (defunct; see Data releases)
License	Proprietary

ABC@home was a volunteer computing project running on the BOINC platform that searched for abc-triples, sets of numbers related to the abc conjecture, one of the best known open problems in number theory. The project was organized by the Mathematical Institute of Leiden University in the Netherlands, together with the Dutch Kennislink science communication institute, and ran from 2006 until it ceased operations in 2015.

ABC@home completed an exhaustive enumeration of all abc-triples with $c<10^{18}$ in 2011, finding 14,482,065 of them, and then searched the range $10^{18}\le c<2^{63}$ non-exhaustively until 2015, bringing the total to 23,827,716 triples. The resulting lists were published as open data and continue to be used in number-theory research.

Background

The abc conjecture, proposed by Joseph Oesterlé and David Masser in 1985, concerns triples of positive, pairwise coprime integers $a$, $b$, and $c$ satisfying

$a+b=c.$

For such a triple, the radical of the product $abc$ is defined as the product of the distinct prime factors of $abc$:

$\mathrm{rad}(abc)=\prod _{p\mid abc}p.$

A triple is called an abc-triple if $\mathrm{rad}(abc)<c$. The quality of an abc-triple is defined as

$q(a,b,c)=\frac{\log c}{\log \mathrm{rad}(abc)}.$

The abc conjecture states that for every $\epsilon >0$, only finitely many abc-triples satisfy $q(a,b,c)>1+\epsilon$. A proof of the conjecture would have far-reaching consequences for number theory, including an asymptotic version of Fermat's Last Theorem and Faltings's theorem.

Two further quantities are used in the project's published data lists: the size of a triple, $\log c/\log 10$ (the number of decimal digits of $c$), and its merit, $(q-1{)}^2\log r\log \log r$ where $r=\mathrm{rad}(abc)$. One triple is said to beat another if it has both a larger size and a larger quality; the list of unbeaten triples gives the best known lower bound on how quickly the limsup of the quality tends to 1. The refined abc conjecture of Robert, Stewart and Tenenbaum predicts that the limsup of the merit is 48.

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.

History

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 Hendrik Lenstra discussed algorithms for enumerating abc-triples. 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.

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. A beta project ran on a separate server from March 2007 to test new application binaries before release.

In April 2011 the project announced that preliminary data for all triples with $c\le 10^{18}$ was available, and that the search had "moved onward to further reaches of the search space". The exhaustive phase had found 14,482,065 triples. A further milestone was announced in September 2012, after which the project warned that far fewer workunits would be available while the data was verified. From 2012 to 2015 the project searched $10^{18}\le c<2^{63}$, adding 9,345,651 triples for a total of 23,827,716; this later phase was not exhaustive.

Free-DC's cumulative statistics list 79,634 users, 146,940 hosts, 1,852 teams and 4,452,051,377 credit for ABC@home. The same totals appear in archived Free-DC captures from 19 September 2015 and 16 October 2026, consistent with a retired project whose statistics no longer change.

Shutdown

The last news item posted by the project, on 17 December 2014, announced a migration to a new machine. Archived copies of the project's own server status page show that the computing pipeline had already stopped earlier: from the snapshot of 25 June 2014 onward, the feeder, transitioner, file deleter, database purge and all three abc_sieve workunit daemons are listed as "Not Running", with zero results ready to send and zero results in progress. Only the web server, upload/download server and scheduler remained running, so the site stayed reachable while no work was issued.

The transitioner backlog reported on that page increases by exactly the elapsed time between snapshots, which places the moment the transitioner stopped at about 18:00 UTC on 1 June 2014. The final archived snapshot, taken on 19 May 2016, still shows the same frozen state: nine workunits awaiting validation, nothing in progress, and a backlog of 17,232 hours. The project's nominal end date is given as 30 November 2015, but the server-side evidence indicates that no new work was distributed after mid-2014.

By April 2017 the abcathome.com domain was redirecting to Bart de Smit's ABC-triples page at Leiden, which states plainly that "At this moment ABC@home is no longer active."

Algorithm

The search avoids testing every pair $(a,b)$ by working with radicals instead. The project's own documentation describes the method as follows.

A number can be a radical only if it is square-free, so a sieve of Eratosthenes variant is used to cross out multiples of prime squares and enumerate all radicals below a bound. All numbers below $N$ having a given radical are then generated by repeatedly multiplying by that radical's prime factors.

The three values $a$, $b$, $c$ are sorted by their radicals into $x$, $y$, $z$ with $r(x)<r(y)<r(z)$. Because the values are coprime, $r(x)r(y)r(z)=\mathrm{rad}(abc)<N$, which gives $r(y)<\sqrt{N}$ and $r(x)<N/r(y{)}^2$. These bounds cut the search space dramatically: only radicals up to $\sqrt{N}$ need to be considered for $y$, and for each choice of $y$ only radicals up to $N/r(y{)}^2$ for $x$. Candidate values of $z$ are then obtained as $x+y$ or $|x-y|$, and only those candidates need to be factored to test the abc condition. Factoring $z$ is the expensive step, but the preceding steps eliminate the vast majority of candidates before it is reached.

The authoritative description of the algorithm, together with the argument for why the resulting list is believed complete and a number of observations about it, is Chapter 7 of Willem Jan Palenstijn's PhD thesis.

Application and platforms

The client application was distributed as ABC finder in the project's early years, reaching version 1.02 by February 2007 on Linux/x86, Linux/x86-64 and Windows/x86. It was later replaced by ABC sieving finder, which reached version 2.10 in August 2010 and was available for Windows (x86 and x86-64), Linux (x86, x86-64 and 32-bit PowerPC) and Mac OS X (Intel, Intel 64-bit and PowerPC). The application was CPU-only; no GPU version was released. Server-side workunit generation, validation and assimilation were handled by the abc_sieve_makework, abc_sieve_validator and abc_sieve_assimilator daemons.

The published minimum requirements were 256 MB of free RAM and 2 MB of free disk space, on Windows, Linux or Mac OS X, with a 64-bit CPU recommended. The implementation of the BOINC client-side search application was addressed separately in a 2011 bachelor thesis by Thijs van Dijk.

Data releases

The project's results were published as two gzipped text files by Bart de Smit at Leiden, and remain the canonical dataset:

• abctriples_below_1018.gz (179 MB) — the complete list of 14,482,065 abc-triples with $c<10^{18}$. Each line holds the values $c$ and $a$ for one triple.
• big_triples.gz (133 MB) — the 9,345,651 additional triples found in $10^{18}\le c<2^{63}$ during 2012–2015. This list is not exhaustive.

Derived lists of high-quality, high-merit and unbeaten triples are published alongside them. The description and analysis of the search algorithm, along with observations about the resulting list of triples, form the final chapter of Palenstijn's thesis.

The abc-triple with the highest known quality, $(2,3^{10}\cdot 109,23^5)$, has a quality of approximately 1.6299 and remains one of the most frequently cited examples in the literature on the conjecture. It was found by Eric Reyssat in 1987 and was not a discovery of the project.

Project personnel

The project's personnel page credited the following people.

• Carl Koppeschaar — initiator
• Hendrik Lenstra — mathematician, initiator
• Willem-Jan Palenstijn — mathematician, programmer
• Bart de Smit — mathematician
• Michiel Vermeulen — project manager
• Hendrik Verhoek — programmer
• Robin Weezepoel — web developer, abcathome.com
• Joris Weimar — web developer, Reken mee met ABC
• Sietske Tacoma — articles for Reken mee met ABC

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 $2^{100}$.

Scientific publications

Papers based on ABC@home data

• Alvarez-Salazar, Elise.(2023).On abc triples of the form (1, c − 1, c). Integers. pp. A64. DOI: 10.5281/zenodo.8283159. Proves general results for constructing infinite families of abc-triples of the form $(1,c-1,c)$, and applies them to a direct statistical analysis of the 45,604 such triples with $c<10^{18}$ found by ABC@home. Published 25 August 2023; the publisher's PDF and volume contents give the paper number as A64. Full text
• Martin, Greg.(2016).abc triples. Functiones et Approximatio Commentarii Mathematici. pp. 145–176. DOI: 10.7169/facm/2016.55.2.2. 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. arXiv:1409.2974
• Palenstijn, Willem Jan.(2014-05-22)."Radicals in Arithmetic".(PhD thesis).Retrieved 2026-11-08. Chapter 7 describes the algorithm used by ABC@home to enumerate abc-triples and reports results from the project; it is cited as the reference for the project's enumeration results by later work analysing the ABC@home data.

Project documentation

• Geuze, Gillien.(2007).Reken mee met ABC. Nieuw Archief voor Wiskunde. pp. 26–30. 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

External links

• ABC@Home statistics — Free-DC project stats, including cumulative users, hosts, teams and credit
• ABC triples — Bart de Smit's data page at Leiden, hosting the project's published triple lists
• Archived ABC@home server status — final captured state of the project server, 19 May 2016
• Radicals in Arithmetic — Palenstijn's PhD thesis, with the ABC@home algorithm and results in Chapter 7