Collatz Conjecture - Revision history

Al Piskun at 11:25, 7 June 2026

2026-06-07T11:25:59Z

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	'''Collatz Conjecture''' was a volunteer distributed computing project built on the [[wikipedia:Berkeley Open Infrastructure for Network Computing\|BOINC]] platform. Based in Wood Dale, Illinois, it is a privately managed project run by administrator Jon Sonntag, dedicated to computationally testing the famous [[wikipedia:Collatz conjecture\|Collatz conjecture]] in mathematics, also widely known as the '''3x+1 problem''' or '''HOTPO''' (Half Or Triple Plus One).<ref name="boinc_wiki">{{cite web \|url=https://boinc.berkeley.edu/wiki/Collatz_Conjecture \|title=Collatz Conjecture \|publisher=BOINC Berkeley \|date=9 May 2013 \|access-date=2026-06-07}}</ref><ref name="project_home">{{cite web \|url=https://boinc.thesonntags.com/collatz/ \|title=Collatz Conjecture project home \|author=Jon Sonntag \|access-date=2026-06-07}}</ref> The project continues work begun by the earlier '''3x+1@home''' BOINC project, which ended in 2008.<ref name="bcteam">{{cite web \|url=https://wiki.bc-team.org/index.php?title=Collatz_Conjecture/en \|title=Collatz Conjecture \|publisher=BC-Wiki \|access-date=2026-06-07}}</ref>		'''[https://web.archive.org/web/20220216091746/https://boinc.thesonntags.com/collatz/ Collatz Conjecture]''' was a volunteer distributed computing project built on the [[wikipedia:Berkeley Open Infrastructure for Network Computing\|BOINC]] platform. Based in Wood Dale, Illinois, it is a privately managed project run by administrator Jon Sonntag, dedicated to computationally testing the famous [[wikipedia:Collatz conjecture\|Collatz conjecture]] in mathematics, also widely known as the '''3x+1 problem''' or '''HOTPO''' (Half Or Triple Plus One).<ref name="boinc_wiki">{{cite web \|url=https://boinc.berkeley.edu/wiki/Collatz_Conjecture \|title=Collatz Conjecture \|publisher=BOINC Berkeley \|date=9 May 2013 \|access-date=2026-06-07}}</ref><ref name="project_home">{{cite web \|url=https://boinc.thesonntags.com/collatz/ \|title=Collatz Conjecture project home \|author=Jon Sonntag \|access-date=2026-06-07}}</ref> The project continues work begun by the earlier '''3x+1@home''' BOINC project, which ended in 2008.<ref name="bcteam">{{cite web \|url=https://wiki.bc-team.org/index.php?title=Collatz_Conjecture/en \|title=Collatz Conjecture \|publisher=BC-Wiki \|access-date=2026-06-07}}</ref>

	== The Collatz Conjecture ==		== The Collatz Conjecture ==

Al Piskun: images and stats

2026-06-07T11:24:33Z

images and stats

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	{{Infobox software		{{Infobox software
	\| name = Collatz Conjecture		\| name = Collatz Conjecture
	\| logo =		\| logo =210pxCollatzFractal.png
	\| logo caption =		\| logo caption =Collatz Conjecture logo
	\| screenshot =		\| screenshot =
	\| caption =		\| caption =
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	'''Collatz Conjecture''' was a volunteer distributed computing project built on the [[wikipedia:Berkeley Open Infrastructure for Network Computing\|BOINC]] platform. Based in Wood Dale, Illinois, it is a privately managed project run by administrator Jon Sonntag, dedicated to computationally testing the famous [[wikipedia:Collatz conjecture\|Collatz conjecture]] in mathematics, also widely known as the '''3x+1 problem''' or '''HOTPO''' (Half Or Triple Plus One).<ref name="boinc_wiki">{{cite web \|url=https://boinc.berkeley.edu/wiki/Collatz_Conjecture \|title=Collatz Conjecture \|publisher=BOINC Berkeley \|date=9 May 2013 \|access-date=2026-06-07}}</ref><ref name="project_home">{{cite web \|url=https://boinc.thesonntags.com/collatz/ \|title=Collatz Conjecture project home \|author=Jon Sonntag \|access-date=2026-06-07}}</ref> The project continues work begun by the earlier '''3x+1@home''' BOINC project, which ended in 2008.<ref name="bcteam">{{cite web \|url=https://wiki.bc-team.org/index.php?title=Collatz_Conjecture/en \|title=Collatz Conjecture \|publisher=BC-Wiki \|access-date=2026-06-07}}</ref>		'''Collatz Conjecture''' was a volunteer distributed computing project built on the [[wikipedia:Berkeley Open Infrastructure for Network Computing\|BOINC]] platform. Based in Wood Dale, Illinois, it is a privately managed project run by administrator Jon Sonntag, dedicated to computationally testing the famous [[wikipedia:Collatz conjecture\|Collatz conjecture]] in mathematics, also widely known as the '''3x+1 problem''' or '''HOTPO''' (Half Or Triple Plus One).<ref name="boinc_wiki">{{cite web \|url=https://boinc.berkeley.edu/wiki/Collatz_Conjecture \|title=Collatz Conjecture \|publisher=BOINC Berkeley \|date=9 May 2013 \|access-date=2026-06-07}}</ref><ref name="project_home">{{cite web \|url=https://boinc.thesonntags.com/collatz/ \|title=Collatz Conjecture project home \|author=Jon Sonntag \|access-date=2026-06-07}}</ref> The project continues work begun by the earlier '''3x+1@home''' BOINC project, which ended in 2008.<ref name="bcteam">{{cite web \|url=https://wiki.bc-team.org/index.php?title=Collatz_Conjecture/en \|title=Collatz Conjecture \|publisher=BC-Wiki \|access-date=2026-06-07}}</ref>

	[[File:Collatz_Conjecture_Loops_all_2.jpg\|thumb\|right\|250px\|Hailstone sequences visualized: the trajectories of several starting numbers before converging to 1. Starting from 27, the sequence climbs as high as 9,232 before descending.]]

	== The Collatz Conjecture ==		== The Collatz Conjecture ==
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	As a striking example, starting from <math>n = 27</math>, the sequence takes 111 steps and climbs as high as 9,232 before descending to 1. Starting from <math>n = 26</math>, by contrast, only 10 steps are needed. There is no obvious pattern to how long a number takes to reach 1 (its '''total stopping time'''), which is part of what makes the problem so difficult.		As a striking example, starting from <math>n = 27</math>, the sequence takes 111 steps and climbs as high as 9,232 before descending to 1. Starting from <math>n = 26</math>, by contrast, only 10 steps are needed. There is no obvious pattern to how long a number takes to reach 1 (its '''total stopping time'''), which is part of what makes the problem so difficult.

	[[File:Collatz5.png\|thumb\|left\|200px\|Graph of the total stopping time for the first 200 starting values. Note the erratic variation — there is no simple formula for predicting how many steps are needed.]]

	The problem goes by several other names in the literature: the '''Ulam conjecture''' (after [[wikipedia:Stanislaw Ulam\|Stanislaw Ulam]]), '''Kakutani's problem''' (after [[wikipedia:Shizuo Kakutani\|Shizuo Kakutani]]), the '''Thwaites conjecture''' (after Sir Bryan Thwaites), and the '''Syracuse problem'''.<ref name="wp_collatz" />		The problem goes by several other names in the literature: the '''Ulam conjecture''' (after [[wikipedia:Stanislaw Ulam\|Stanislaw Ulam]]), '''Kakutani's problem''' (after [[wikipedia:Shizuo Kakutani\|Shizuo Kakutani]]), the '''Thwaites conjecture''' (after Sir Bryan Thwaites), and the '''Syracuse problem'''.<ref name="wp_collatz" />
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	== Computational verification ==		== Computational verification ==
			[[File:All Collatz sequences of a length inferior to 20.svg\|thumb\|360x360px\|All Collatz sequences of a length inferior to 20]]
	Although no general proof exists, computers have verified the conjecture for enormous ranges of integers. As of the most recent exhaustive computations, the conjecture has been confirmed to hold for all positive integers up to <math>2^{68} \approx 2.95 \times 10^{20}</math> (roughly 295 quintillion), with no counterexample found.<ref name="tao2019">{{cite arXiv \|last=Tao \|first=Terence \|title=Almost all orbits of the Collatz map attain almost bounded values \|eprint=1909.03562 \|year=2019}}</ref><ref name="springer2025">{{cite journal \|title=Computational verification of the Collatz conjecture up to 2^71 \|journal=The Journal of Supercomputing \|year=2025 \|doi=10.1007/s11227-025-07337-0}}</ref>		Although no general proof exists, computers have verified the conjecture for enormous ranges of integers. As of the most recent exhaustive computations, the conjecture has been confirmed to hold for all positive integers up to <math>2^{68} \approx 2.95 \times 10^{20}</math> (roughly 295 quintillion), with no counterexample found.<ref name="tao2019">{{cite arXiv \|last=Tao \|first=Terence \|title=Almost all orbits of the Collatz map attain almost bounded values \|eprint=1909.03562 \|year=2019}}</ref><ref name="springer2025">{{cite journal \|title=Computational verification of the Collatz conjecture up to 2^71 \|journal=The Journal of Supercomputing \|year=2025 \|doi=10.1007/s11227-025-07337-0}}</ref>

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	=== Technical details ===		=== Technical details ===
			[[File:Collatz Fractal.jpg\|thumb\|360x360px\|Collatz Fractal]]
	The project makes use of the '''parity sequence optimization''', a well-known algorithmic shortcut that condenses multiple Collatz steps into a single operation by analyzing the parity (odd/even) pattern of consecutive iterations, allowing the software to skip redundant even-division steps and test more integers per unit time.<ref name="tsbt">{{cite web \|url=https://tsbt.co.uk/forum/viewtopic.php?t=9264 \|title=Collatz Conjecture New Project Details \|publisher=The Scottish BOINC Team \|access-date=2026-06-07}}</ref>		The project makes use of the '''parity sequence optimization''', a well-known algorithmic shortcut that condenses multiple Collatz steps into a single operation by analyzing the parity (odd/even) pattern of consecutive iterations, allowing the software to skip redundant even-division steps and test more integers per unit time.<ref name="tsbt">{{cite web \|url=https://tsbt.co.uk/forum/viewtopic.php?t=9264 \|title=Collatz Conjecture New Project Details \|publisher=The Scottish BOINC Team \|access-date=2026-06-07}}</ref>

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	== Scientific context ==		== Scientific context ==

			[[File:Lothar_Collatz.jpg\|thumb\|right\|180px\|Lothar Collatz (1910–1990), the German mathematician who proposed the 3x+1 conjecture in 1937.]]

	=== The 3x+1 problem in the literature ===		=== The 3x+1 problem in the literature ===
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	The Collatz Conjecture BOINC project is not the only computational effort aimed at the problem. The '''yoyo@home''' BOINC project independently checked all positive integers up to <math>10^{20} \approx 2^{66.4}</math> for convergence in 2017, using CPU-based methods with roughly 1,000 volunteers.<ref name="springer2025" /> A separate standalone computation by Tomas Oliveira e Silva in 2009 verified the conjecture up to approximately <math>5.76 \times 10^{18} \approx 2^{62.3}</math>.<ref name="springer2025" /> Collatz Conjecture BOINC focuses specifically on testing much larger numbers in ranges above what exhaustive sequential verification has covered.		The Collatz Conjecture BOINC project is not the only computational effort aimed at the problem. The '''yoyo@home''' BOINC project independently checked all positive integers up to <math>10^{20} \approx 2^{66.4}</math> for convergence in 2017, using CPU-based methods with roughly 1,000 volunteers.<ref name="springer2025" /> A separate standalone computation by Tomas Oliveira e Silva in 2009 verified the conjecture up to approximately <math>5.76 \times 10^{18} \approx 2^{62.3}</math>.<ref name="springer2025" /> Collatz Conjecture BOINC focuses specifically on testing much larger numbers in ranges above what exhaustive sequential verification has covered.

	~~[[File:Lothar_Collatz.jpg\|thumb\|right\|180px\|Lothar Collatz (1910–1990), the German mathematician who proposed the 3x+1 conjecture in 1937.]]~~

	== See also ==		== See also ==

Al Piskun: first light

2026-06-07T11:12:33Z

first light

New page

{{Infobox software
| name = Collatz Conjecture
| logo =
| logo caption =
| screenshot =
| caption =
| description = Collatz Conjecture is an active Mathematics BOINC project based in Wood Dale, Illinois, testing the 3x+1 conjecture using CPU and GPU volunteer computing, managed privately by Jon Sonntag.

| status = Inactive
| category = Mathematics
| compute = CPU & GPU
| dependencies = OpenCL driver (GPU); Microsoft Visual C++ Redistributable (CPU)

| developer = Jon Sonntag
| author = Jon Sonntag
| sponsor = Self-funded (private)
| maintainer = Jon Sonntag
| released = {{Start date and age|2009|01|06}}
| completed =
| discontinued =
| repository =

| programming language = C, C++ (CPU); OpenCL (GPU)
| operating system = Windows, Linux, macOS, Android
| size =

| stats as of = {{Start date and age|2022|01|19}}
| average performance = 13254678.71 GigaFLOPS
| active users = 1949
| total users = 67352
| active hosts = 4887
| total hosts = 433270

| rac =
| credit per day =
| gpu performance =
| cpu performance =

| website = {{URL|https://boinc.thesonntags.com/collatz/}}
| license = Proprietary (closed source)
}}

'''Collatz Conjecture''' was a volunteer distributed computing project built on the [[wikipedia:Berkeley Open Infrastructure for Network Computing|BOINC]] platform. Based in Wood Dale, Illinois, it is a privately managed project run by administrator Jon Sonntag, dedicated to computationally testing the famous [[wikipedia:Collatz conjecture|Collatz conjecture]] in mathematics, also widely known as the '''3x+1 problem''' or '''HOTPO''' (Half Or Triple Plus One).<ref name="boinc_wiki">{{cite web |url=https://boinc.berkeley.edu/wiki/Collatz_Conjecture |title=Collatz Conjecture |publisher=BOINC Berkeley |date=9 May 2013 |access-date=2026-06-07}}</ref><ref name="project_home">{{cite web |url=https://boinc.thesonntags.com/collatz/ |title=Collatz Conjecture project home |author=Jon Sonntag |access-date=2026-06-07}}</ref> The project continues work begun by the earlier '''3x+1@home''' BOINC project, which ended in 2008.<ref name="bcteam">{{cite web |url=https://wiki.bc-team.org/index.php?title=Collatz_Conjecture/en |title=Collatz Conjecture |publisher=BC-Wiki |access-date=2026-06-07}}</ref>

[[File:Collatz_Conjecture_Loops_all_2.jpg|thumb|right|250px|Hailstone sequences visualized: the trajectories of several starting numbers before converging to 1. Starting from 27, the sequence climbs as high as 9,232 before descending.]]

== The Collatz Conjecture ==

The Collatz conjecture is one of the most famous and enduring unsolved problems in mathematics. It was first proposed by German mathematician [[wikipedia:Lothar Collatz|Lothar Collatz]] (6 July 1910 – 26 September 1990) in 1937, two years after receiving his doctorate from the University of Berlin under Alfred Klose.<ref name="mactutor">{{cite web |url=https://mathshistory.st-andrews.ac.uk/Biographies/Collatz/ |title=Lothar Collatz (1910–1990) |publisher=MacTutor History of Mathematics |access-date=2026-06-07}}</ref> Despite its deceptively elementary statement, the conjecture has resisted all attempts at a general proof for nearly ninety years.

=== Statement of the conjecture ===

Define the Collatz function <math>f</math> on the positive integers as follows:

:<math>f(n) = \begin{cases} n/2 & \text{if } n \equiv 0 \pmod{2} \\ 3n + 1 & \text{if } n \equiv 1 \pmod{2} \end{cases}</math>

Given any starting positive integer <math>n</math>, one forms a sequence by repeatedly applying <math>f</math>:

:<math>n,\ f(n),\ f^2(n),\ f^3(n),\ \ldots</math>

The '''Collatz conjecture''' asserts that no matter which positive integer <math>n</math> is chosen, this sequence will eventually reach the number 1.<ref name="wp_collatz">{{cite web |url=https://en.wikipedia.org/wiki/Collatz_conjecture |title=Collatz conjecture |publisher=Wikipedia |access-date=2026-06-07}}</ref>

A common shortcut collapses the two steps applied to an odd number into one, giving the equivalent '''Syracuse map''':

:<math>T(n) = \frac{3n+1}{2^{v_2(3n+1)}}</math>

where <math>v_2</math> denotes the 2-adic valuation (the largest power of 2 dividing the argument). This form is used in several efficient computational implementations.

=== Hailstone sequences ===

Because the values in a Collatz sequence rise and fall unpredictably before eventually reaching 1, they are often called '''hailstone sequences''' or '''hailstone numbers''', a name that evokes the way real hailstones tumble up and down inside a thundercloud before finally falling to the ground.<ref name="wp_collatz" /> They are also sometimes called '''wondrous numbers'''.

As a striking example, starting from <math>n = 27</math>, the sequence takes 111 steps and climbs as high as 9,232 before descending to 1. Starting from <math>n = 26</math>, by contrast, only 10 steps are needed. There is no obvious pattern to how long a number takes to reach 1 (its '''total stopping time'''), which is part of what makes the problem so difficult.

[[File:Collatz5.png|thumb|left|200px|Graph of the total stopping time for the first 200 starting values. Note the erratic variation — there is no simple formula for predicting how many steps are needed.]]

The problem goes by several other names in the literature: the '''Ulam conjecture''' (after [[wikipedia:Stanislaw Ulam|Stanislaw Ulam]]), '''Kakutani's problem''' (after [[wikipedia:Shizuo Kakutani|Shizuo Kakutani]]), the '''Thwaites conjecture''' (after Sir Bryan Thwaites), and the '''Syracuse problem'''.<ref name="wp_collatz" />

=== Reactions from mathematicians ===

The difficulty of the conjecture is legendary. [[wikipedia:Paul Erdős|Paul Erdős]], one of the most prolific mathematicians of the twentieth century, is reported to have said: ''Mathematics may not be ready for such problems.''<ref name="wp_collatz" /> [[wikipedia:Jeffrey Lagarias|Jeffrey Lagarias]], who has compiled the definitive research bibliography on the problem, wrote in 2010 that the conjecture is ''an extraordinarily difficult problem, completely out of reach of present day mathematics''.<ref name="lagarias2010">{{cite book |last=Lagarias |first=Jeffrey C. |title=The Ultimate Challenge: The 3x+1 Problem |publisher=American Mathematical Society |year=2010 |isbn=978-0-8218-4940-8}}</ref>

== Computational verification ==

Although no general proof exists, computers have verified the conjecture for enormous ranges of integers. As of the most recent exhaustive computations, the conjecture has been confirmed to hold for all positive integers up to <math>2^{68} \approx 2.95 \times 10^{20}</math> (roughly 295 quintillion), with no counterexample found.<ref name="tao2019">{{cite arXiv |last=Tao |first=Terence |title=Almost all orbits of the Collatz map attain almost bounded values |eprint=1909.03562 |year=2019}}</ref><ref name="springer2025">{{cite journal |title=Computational verification of the Collatz conjecture up to 2^71 |journal=The Journal of Supercomputing |year=2025 |doi=10.1007/s11227-025-07337-0}}</ref>

A notable milestone is Terence Tao's 2019 paper "Almost all orbits of the Collatz map attain almost bounded values," which proved that for ''any'' function <math>f : \mathbb{N} \to \mathbb{R}</math> with <math>\lim_{N \to \infty} f(N) = +\infty</math>, the minimum element of the Collatz orbit of <math>N</math> satisfies <math>\mathrm{Col}_{\min}(N) \leq f(N)</math> for almost all <math>N</math> in the sense of logarithmic density.<ref name="tao2019" /> While not a full proof, this was widely regarded as the most significant theoretical advance on the problem in decades.

A more recent effort, published in ''The Journal of Supercomputing'' (2025), pushed the verified limit to <math>2^{71}</math> using GPU-accelerated algorithms distributed across several European supercomputers, with a total acceleration of <math>1{,}335\times</math> compared to baseline CPU implementations and the discovery of four new path records along the way.<ref name="springer2025" />

== About the BOINC project ==

=== Origins ===

The Collatz Conjecture BOINC project was launched in 2009 by Jon Sonntag of Wood Dale, Illinois, continuing work left off by the earlier '''3x+1@home''' BOINC project, which had ended in 2008.<ref name="bcteam" /><ref name="boinc_australia">{{cite web |url=http://forum.boinc-australia.net/index.php?board=136.0 |title=Collatz Conjecture forum |publisher=BOINC@Australia |access-date=2026-06-07}}</ref> The project is self-funded and privately operated, without university or government backing — something its administrator has cited as a point of pride, demonstrating that a single motivated individual working part-time can run a competitive BOINC project.<ref name="boinc_ru">{{cite web |url=https://boinc.ru/persona-grata/o-proekte-collatz-conjecture-iz-pervyh-ruk-intervyu-s-rukovoditelem-proekta/ |title=About project Collatz Conjecture first-hand (Interview with the Project Manager) |publisher=BOINC.RU |date=September 2011 |access-date=2026-06-07}}</ref>

The project's stated goal is to search for a '''counterexample''' to the Collatz conjecture — a starting number whose sequence never reaches 1. Since all numbers below a given threshold have been verified elsewhere, the project focuses on progressively larger ranges, having started from around <math>2^{71}</math>. As Sonntag put it in a 2011 interview, at peak activity volunteers were collectively testing roughly 40 quadrillion numbers per day, far exceeding initial expectations.<ref name="boinc_ru" />

=== GPU computing and early milestones ===

Collatz Conjecture holds a notable place in BOINC history as one of the first projects to support AMD/ATI graphics cards without requiring volunteers to supply a custom <code>app_info.xml</code> file. This was made possible by collaborations with BOINC community developers "Crunch3r" (who contributed BOINC client work) and "Gipsel" (who wrote hand-optimized GPU assembly code for ATI hardware).<ref name="boinc_ru" /> These contributions, and Gipsel's hand-written GPU assembly in particular, demonstrated how much faster GPU-based computation could be compared to CPU-only methods.

At the time the project launched in 2009, [[wikipedia:MilkyWay@home|MilkyWay@home]] was essentially the only other BOINC project supporting ATI GPU computing, and it suffered from frequent outages. Sonntag explicitly wanted to give GPU volunteers an alternative, and also to push the frontier of GPU support within BOINC.<ref name="boinc_ru" />

=== Technical details ===

The project makes use of the '''parity sequence optimization''', a well-known algorithmic shortcut that condenses multiple Collatz steps into a single operation by analyzing the parity (odd/even) pattern of consecutive iterations, allowing the software to skip redundant even-division steps and test more integers per unit time.<ref name="tsbt">{{cite web |url=https://tsbt.co.uk/forum/viewtopic.php?t=9264 |title=Collatz Conjecture New Project Details |publisher=The Scottish BOINC Team |access-date=2026-06-07}}</ref>

The application is available for multiple platforms and computing devices:

{| class="wikitable"
! Platform !! Supported devices
|-
| Windows || CPU, NVIDIA GPU (CUDA), AMD GPU (OpenCL), Intel GPU (OpenCL)
|-
| Linux || CPU, NVIDIA GPU, AMD GPU, Intel GPU
|-
| macOS || CPU, GPU (OpenCL)
|-
| Android || CPU
|}

For GPU computing, minimum requirements include an AMD Radeon HD 5000 or later with OpenCL 1.1 support, an NVIDIA GeForce 8400 GS or later with a recent NVIDIA driver, or an Intel HD 2500 or later with Intel's OpenCL runtime.<ref name="tsbt" />

Because spammers created large numbers of bogus accounts in the project's early years, invite codes are required to register. The code changes periodically and is available from the project administrator.<ref name="anandtech">{{cite web |url=https://forums.anandtech.com/threads/about-collatz-conjecture.2504696/ |title=About Collatz Conjecture |publisher=AnandTech Forums |access-date=2026-06-07}}</ref>

=== 2018 hardware failure ===

In early 2018 the project suffered a complete hardware failure, and nearly all data was lost. Credit totals were eventually recovered and restored to participants. Shortly after recovering, new achievement badges were introduced for the project.<ref name="bcteam" />

=== Reliability and downtime ===

As a privately hosted project, Collatz Conjecture has experienced periodic outages over the years, typically related to server hardware, dynamic DNS configuration, or code-signing key renewal. The project's forum and server share the same infrastructure, which can complicate communication during outages. When such events occur, the administrator has generally addressed them promptly via BOINC community forums.<ref name="boinc_forum_down">{{cite web |url=https://isaac.ssl.berkeley.edu/forum_thread.php?id=12949 |title=Collatz Conjecture Down? |publisher=BOINC message boards |date=14 May 2019 |access-date=2026-06-07}}</ref>

=== Credit and performance ===

Collatz Conjecture is well known within the BOINC community for issuing substantial amounts of BOINC credit relative to computation time, particularly for GPU tasks. Community benchmarks have indicated that a high-end GPU can generate on the order of millions of BOINC cobblestones per day on this project. This generosity in credit issuance has attracted both enthusiastic participants and some controversy in BOINC forums.<ref name="anandtech" />

== Scientific context ==

=== The 3x+1 problem in the literature ===

The Collatz conjecture sits at the intersection of elementary number theory and computational mathematics. The authoritative research compilation is Jeffrey Lagarias's annotated bibliography, ''The 3x+1 Problem: An Annotated Bibliography'' (2006, updated), which lists hundreds of papers touching on the problem from every angle.<ref name="lagarias2010" />

Key theoretical results prior to Tao's 2019 paper include Krasikov and Lagarias (2003), who proved that the count of integers in <math>[1, x]</math> satisfying the conjecture grows at least as fast as <math>x^{0.84}</math> for sufficiently large <math>x</math>.<ref name="tao2019" /> Conway (1972) showed that a slight variant of the <math>qn+1</math> problem is undecidable, suggesting the original problem sits at the very edge of the decidability threshold.<ref name="wp_collatz" />

=== Related volunteer computing efforts ===

The Collatz Conjecture BOINC project is not the only computational effort aimed at the problem. The '''yoyo@home''' BOINC project independently checked all positive integers up to <math>10^{20} \approx 2^{66.4}</math> for convergence in 2017, using CPU-based methods with roughly 1,000 volunteers.<ref name="springer2025" /> A separate standalone computation by Tomas Oliveira e Silva in 2009 verified the conjecture up to approximately <math>5.76 \times 10^{18} \approx 2^{62.3}</math>.<ref name="springer2025" /> Collatz Conjecture BOINC focuses specifically on testing much larger numbers in ranges above what exhaustive sequential verification has covered.

[[File:Lothar_Collatz.jpg|thumb|right|180px|Lothar Collatz (1910–1990), the German mathematician who proposed the 3x+1 conjecture in 1937.]]

== See also ==

* [[wikipedia:BOINC|BOINC]]
* [[wikipedia:Collatz conjecture|Collatz conjecture]]
* [[wikipedia:Volunteer computing|Volunteer computing]]
* [[wikipedia:MilkyWay@home|MilkyWay@home]]
* [[yoyo@home]]
* [[PrimeGrid]]

== References ==

{{Reflist}}

== External links ==

* {{URL|https://web.archive.org/web/20220216091746/https://boinc.thesonntags.com/collatz/}} – Web Archive of the project
* {{URL|https://en.wikipedia.org/wiki/Collatz_conjecture}} – Wikipedia: Collatz conjecture
* {{URL|https://arxiv.org/abs/1909.03562}} – Terence Tao (2019): "Almost all orbits of the Collatz map attain almost bounded values"
* {{URL|https://boinc.berkeley.edu/wiki/Collatz_Conjecture}} – BOINC project listing

[[Category:BOINC projects]]
[[Category:Mathematics BOINC projects]]
[[Category:Active BOINC projects]]
[[Category:GPU computing projects]]
[[Category:Number theory]]