SPT

SPT (Symmetric Prime Tuples)
SPT project logo
Project
Status	Active
Category	Mathematics / Number Theory
Compute	CPU
Development
Developer	Demis
Author	Natalia Makarova
Sponsor	Natalia Makarova, Alex Belyshev, Tomáš Brada
Maintainer	Natalia Makarova, Alex Belyshev, Tomáš Brada
Initial release	2023
Software
Written in	C, C++
Operating system	Windows, Linux
BOINC statistics
Stats as of	23 May 2026
Active users	0
Total users	192
Active hosts	0
Total hosts	1,026
Metadata
Website	https://boinc.termit.me/adsl/

BOINC, the Berkeley Open Infrastructure for Network Computing, is the platform that powers SPT.

SPT (Symmetric Prime Tuples) is a BOINC-based volunteer computing project dedicated to researching symmetric prime k-tuples of consecutive primes, a topic in number theory. Volunteers donate idle CPU time on their personal computers to help systematically search for symmetric prime k-tuples, advancing an open mathematical problem first formally presented by mathematician Natalia Makarova. The project is hosted at boinc.termit.me/adsl/ and serves as the direct continuation of an earlier lineage of BOINC projects pursuing the same mathematical goal.

Background and History

The Ulam spiral, a visualisation of the distribution of prime numbers. SPT investigates a specific structural property of consecutive primes: whether they can be arranged in symmetric patterns.

The search for symmetric prime tuples has a rich history within the volunteer computing community. The research lineage traces back to Stop@Home, an early BOINC project that searched for symmetric prime sequences. Stop@Home achieved a notable result: in April 2017, the project's volunteers discovered a minimal symmetric 17-tuple with starting prime $p=159,067,808,851,610,411$ and offsets 0, 42, 60, 96, 102, 186, 210, 240, 246, 252, 282, 306, 390, 396, 432, 450, 492.

When Stop@Home was retired, the work was taken up as a subproject by T.Brada Experimental Grid (TBEG), a multi-application BOINC project run by Tomáš Brada. The TBEG symmetric prime tuples subproject was announced on 18 October 2019 after multiple closed tests, and was described explicitly as "a continuation of deceased project Stop@home." The subproject extended the search beyond symmetric prime tuples to also cover twin prime tuples, symmetric twin prime tuples, and prime tuple gaps, and extended the search interval accordingly. The TBEG subproject was available for both Linux and Windows, and ran until the project's closure in 2022.

After TBEG went offline, Natalia Makarova and her colleagues established the standalone SPT project in 2023 to continue this research without interruption. The project was configured by a developer known as Demis, who adapted the original source code from Tomáš Brada's repository for the new platform. A parallel branch of the same research also runs within Gerasim@Home under the application name "Get Symmetrical Tuples", which employs a different algorithm targeting only odd-length tuples.

Goal

The primary goal of SPT is to continue the work of the T. Brada Experimental Grid project by systematically searching for symmetric prime k-tuples of consecutive primes. The project seeks to find solutions with minimal diameter and minimal value of the starting prime $p$ for various values of $k$, and to discover solutions for values of $k$ for which none are yet known. All results are published publicly in the Prime Tuple Database, hosted on the T. Brada Experimental Grid website.

Mathematical Background

The SPT application is based on Problem 62. Symmetric k-tuples of consecutive primes, a mathematical puzzle submitted by Natalia Makarova to Carlos Rivera's Prime Puzzles and Problems Connection website.

Definition 1: Prime k-tuple

A prime k-tuple is a finite collection of values

$(p+a_1,p+a_2,p+a_3,\dots ,p+a_k)$

where $p,p+a_1,p+a_2,\dots ,p+a_k$ are prime numbers, and $(a_1,a_2,a_3,\dots ,a_k)$ form a pattern. Typically the first value in the pattern is 0 and the remaining values are distinct positive even numbers.

In SPT, the search is further restricted to the case where $p+a_1,p+a_2,\dots ,p+a_k$ are consecutive primes, meaning there are no primes between any two adjacent elements of the tuple.

Definition 2: Symmetric k-tuple (even k)

For even $k$, a k-tuple

$(p+a_1,p+a_2,\dots ,p+a_{k/2},p+a_{k/2+1},\dots ,p+a_{k-1},p+a_k)$

is called symmetric if the following condition is satisfied:

$a_1+a_k=a_2+a_{k-1}=a_3+a_{k-2}=\cdots =a_{k/2}+a_{k/2+1}$

That is, the offsets mirror each other around the centre of the tuple.

Example: symmetric 8-tuple

The tuple $(17+0,17+2,17+6,17+12,17+14,17+20,17+24,17+26)$ satisfies the symmetry condition, since $0+26=2+24=6+20=12+14=26$. In shorthand notation this is written as:

17: 0, 2, 6, 12, 14, 20, 24, 26

Definition 3: Symmetric k-tuple (odd k)

For odd $k$, a k-tuple is called symmetric if:

$a_1+a_k=a_2+a_{k-1}=\cdots =a_{(k-1)/2}+a_{(k-1)/2+2}=2a_{(k-1)/2+1}$

The middle element $a_{(k-1)/2+1}$ is exactly the average of its flanking pair, making it the literal centre of symmetry.

Example: symmetric 5-tuple

18713: 0, 6, 18, 30, 36

Here the middle offset is 18, and $0+36=6+30=2\times 18=36$, confirming symmetry.

Definition 4: Diameter

The diameter $d$ of a k-tuple is the difference between its largest and smallest offsets:

$d=a_k-a_1$

For the 8-tuple example above, $d=26-0=26$. Known solutions with minimal diameter and minimal starting prime $p$ for $k=2,4,6,8$ are catalogued in OEIS sequence A081235.

Open Questions

The project is working to answer two main questions posed by Makarova:

1. Find solutions with a minimal diameter and minimal starting prime $p$ for $10<k<17$, and for $k=18,20,22,24$.
2. Find solutions (minimal or otherwise) for the remaining even and odd values of $k$.

As of the project's current knowledge, no solutions have been found for $k=17,19,21,23$.

Known Solutions

The Sieve of Eratosthenes, the classical prime-finding algorithm underlying the modern primesieve library used by SPT.

The table below summarises the best known solutions for various values of $k$, as published on the project's reference page. Solutions marked "not minimal" indicate that a smaller starting prime $p$ or smaller diameter may still exist. Results were discovered across Stop@Home, TBEG, manual computation by contributors such as J. Wroblewski and two anonymous Russian contributors, and the ongoing SPT project.

k	Status	Starting prime p	Offsets
5	minimal d, p	18,713	0, 6, 18, 30, 36
7	minimal d, p	12,003,179	0, 12, 18, 30, 42, 48, 60
9	minimal d, p	1,480,028,129	0, 12, 24, 30, 42, 54, 60, 72, 84
10	minimal d, p	13	0, 4, 6, 10, 16, 18, 24, 28, 30, 34
11	possible minimal	660,287,401,247,651	0, 6, 30, 42, 60, 66, 72, 90, 102, 126, 132
12	not minimal	137	0, 2, 12, 14, 20, 26, 30, 36, 42, 44, 54, 56
13	not minimal	5,348,080,416,833,699	0, 12, 30, 42, 48, 72, 90, 108, 132, 138, 150, 168, 180
14	not minimal	19,636,011,281,690,651	0, 2, 8, 12, 18, 26, 30, 38, 42, 50, 56, 60, 66, 68
15	not minimal	5,348,080,416,833,681	0, 18, 30, 48, 60, 66, 90, 108, 126, 150, 156, 168, 186, 198, 216
16	not minimal	19,636,011,281,690,647	0, 4, 6, 12, 16, 22, 30, 34, 42, 46, 54, 60, 64, 70, 72, 76
17	minimal (J. Wroblewski)	258,406,392,900,394,343,851	0, 12, 30, 42, 60, 72, 78, 102, 120, 138, 162, 168, 180, 198, 210, 228, 240
17	minimal (Stop@Home, April 2017)	159,067,808,851,610,411	0, 42, 60, 96, 102, 186, 210, 240, 246, 252, 282, 306, 390, 396, 432, 450, 492
18	not minimal	49,549,273,441,123	0, 4, 24, 40, 46, 54, 58, 66, 70, 84, 88, 96, 100, 108, 114, 130, 150, 154
20	not minimal	11,785,542,108,641,839	0, 4, 10, 18, 24, 30, 52, 70, 72, 84, 118, 130, 132, 150, 172, 178, 184, 192, 198, 202
22	not minimal	18,620,445,306,703,861	0, 10, 36, 46, 66, 76, 82, 96, 102, 130, 136, 162, 168, 196, 202, 216, 222, 232, 252, 262, 288, 298
24	not minimal	22,930,603,692,243,271	0, 70, 76, 118, 136, 156, 160, 178, 202, 222, 238, 250, 378, 390, 406, 426, 450, 468, 472, 492, 510, 552, 558, 628

Notable contributor results include a separate set of minimal-diameter solutions discovered by two anonymous Russian mathematicians (submitted via [email protected] and [email protected]) for $k=11,12,13,14,16$, shared on the primepuzzles.net discussion page. Independently, J. Wroblewski found minimal-diameter solutions for $k=15,17,18,20$ during a contest organised by Makarova and her colleague S. Tognon in 2015.

Methods

Prime Matrix Construction

The SPT application is computationally demanding by design. Each work unit requires approximately 1.3 GB of RAM, nearly all of which is consumed during the construction of a matrix of prime numbers. This prime matrix is built using primesieve, an open-source C/C++ library by Kim Walisch that generates prime numbers using the segmented Sieve of Eratosthenes with wheel factorization. The library detects the CPU's L1 and L2 cache sizes and allocates its data structures accordingly, making it well-suited to the large prime searches required by SPT.

The specific version of primesieve used in SPT is pinned to commit 2b2c4a5.

Search Algorithm

Once the prime matrix is constructed, the tuple-search code operates on only 5 to 6 MB of memory and does not grow during execution. The search algorithm was originally written by Tomáš Brada (available at his GitHub repository) and subsequently adapted for the SPT BOINC project by Demis (at the SPT GitHub repository).

Memory Growth Over Time

As the project searches progressively larger prime numbers, the prime matrix must grow to cover the expanded search range. When SPT first launched, the matrix for a single task was approximately 620 MB; it has since grown to 1.3 GB and will continue to increase slowly as the numbers under investigation become larger.

Work Unit Timing

A single work unit takes on average between 45 minutes and 1.5 hours to complete, though significant variation exists. Some machines complete tasks in as little as 20 minutes, while slower or memory-constrained machines may take up to 240 minutes. The project server allows up to 6 days for a task to be returned before it is reassigned to another volunteer.

The FAQ cautions volunteers about task overcommitment. A 4-core machine that accepts 1,000 tasks at 1 hour per task per core can process at most $4\times 24\times 6=576$ tasks in the 6-day deadline window, leaving $1000-576=424$ tasks to time out and be reassigned by the server automatically.

Resource Management Recommendations

Because SPT tasks are memory-intensive, the project FAQ provides guidance for users on setting appropriate resource limits. For a machine with 4 GB of RAM and 4 logical cores (2 physical with hyperthreading enabled), one core should be reserved for the operating system. Running 3 simultaneous SPT tasks would require

$3\times 1.3\text{GB}=3.9\text{GB}$

of RAM, which is likely to force the system to use swap memory and cause severe slowdown. The project therefore recommends limiting concurrent tasks to 2 on such hardware, corresponding to a 50% CPU limit in BOINC Manager.

Advanced users may configure a local app_config.xml file placed in the project directory (/projects/boinc.termit.me_adsl/) to cap the number of simultaneously running tasks:

<syntaxhighlight lang="xml"> <app_config>

   <project_max_concurrent>8</project_max_concurrent>

</app_config> </syntaxhighlight>

The value should be set to reflect available RAM, where each task requires approximately 1.3 GB.

Project Team

SPT is maintained by a small international team of mathematicians and developers:

• Demis - developer who configured the SPT BOINC server and adapted the application code for the project.
• Natalia Makarova - project scientist and originator of the mathematical problem underlying SPT; author of the problem formulation on primepuzzles.net.
• Alex Belyshev - project co-maintainer.
• Tomáš Brada - author of the original symmetric prime tuple search code used as the basis for SPT, and former operator of the T. Brada Experimental Grid.

Scientific Results

All results produced by the SPT project are made publicly available through the Prime Tuple Database, hosted on the T. Brada Experimental Grid website. Raw result archives are downloadable directly from the project server in several formats:

• Results as-is from the table
• Results with offsets converted from ofs to dn
• Results converted to an

The project's findings contribute to OEIS sequences related to prime k-tuples, including A081235 (minimal symmetric prime tuples) and A055380 (diameter of prime k-tuples).

Project Statistics

As of 23 May 2026, the SPT server status reports 192 registered users with credit and 1,026 registered hosts. All server infrastructure components (download server, upload server, scheduler, feeder, transitioner, and file deleter) are reported as running normally. The project is currently in a period of low activity, with no tasks in progress or ready to send as of the statistics date.

Participating

To join the SPT project, download and install the BOINC client and attach to the project using the URL:

https://boinc.termit.me/adsl/

The project supports Windows and Linux operating systems. Tasks run on CPU only; there is no GPU support. Given the 1.3 GB RAM requirement per task, users with limited memory are encouraged to consult the project FAQ before configuring BOINC resource limits.

See Also

• Prime k-tuple
• Sieve of Eratosthenes
• Volunteer computing
• BOINC
• Gerasim@Home
• T.Brada Experimental Grid
• PrimeGrid

External Links

• SPT Official Website
• SPT FAQ
• SPT Server Status
• Prime Tuple Database
• Problem 62 - Symmetric k-tuples of consecutive primes (primepuzzles.net)
• OEIS A081235 - Minimal symmetric prime tuples
• OEIS A055380 - Diameter of prime k-tuples
• SPT source code (GitHub)
• Original T. Brada source code (GitHub)
• primesieve library (GitHub)