ABC@home

ABC@home
ABC@home project logo
Project
Status	Completed
Category	Mathematics
Compute	CPU
Requires	None
Development
Developer	Mathematical Institute, Leiden University
Author	Bart de Smit
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	Cross-platform
BOINC statistics
Stats as of	March 15, 2011  (15 years ago)
Performance	~10 teraflops
Active users	7,300
Analytics
CPU performance	~10 teraflops
Metadata
Website	http://www.abcathome.com/
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.

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.

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. By March 2011, the project had more than 7,300 active participants from 114 countries, with a combined BOINC credit total exceeding 2.9 billion and an aggregate throughput of roughly 10 teraflops.

In 2011, the project reached a major milestone: it completed an exhaustive search finding all abc-triples $(a,b,c)$ with $c<10^{18}$, a total of 14,482,065 triples. The project continued searching for larger triples with $10^{18}\le c<2^{63}$ 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.

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. 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. The implementation of the BOINC client-side search application was separately addressed in a bachelor thesis by Thijs van Dijk in 2011.

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.

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 $2^{100}$.

Publications

Papers based on ABC@home data

• Martin, Greg.(2016).abc triples. Functiones et Approximatio Commentarii Mathematici. pp. 145–176. 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
• Alvarez-Salazar, Elise.(2023).On abc triples of the form (1, c − 1, c). arXiv preprint. 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.

Project documentation

• Palenstijn, Willem Jan.(2014-05-22).Radicals in Arithmetic. Leiden University Scholarly Publications. Universiteit Leiden. Retrieved 2026-07-18. Chapter 5 describes the algorithm used by ABC@home to enumerate abc-triples and reports results from the project.
• 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