PRIVATE GFN SERVER

BOINC project PRIVATE GFN SERVER is a volunteer computing project researching numbers of the form

PRIVATE GFN SERVER
Project
Status	Active
Category	Mathematics / Number theory
Compute	CPU & GPU
Development
Developer	stream
Author	stream
Initial release	c. 2016
Repository	https://github.com/galloty/genefer20
Software
Operating system	Windows, Linux
BOINC statistics
Stats as of	May 19, 2026  (0 years ago)
Performance	111809.79 GigaFLOPS
Active users	127
Total users	559
Active hosts	596
Total hosts	13,072
Metadata
Website	http://boincvm.proxyma.ru:30080/test4vm/

$a^{2^n}+b^{2^n}$

with a, b any coprime integers, a > b > 0 — Generalized Fermat numbers.

Background: Generalized Fermat numbers

A classical Fermat number has the form $F_n=2^{2^n}+1$. Pierre de Fermat (1601–1665) conjectured that all such numbers are prime; Leonhard Euler disproved this in 1732 by showing $F_5=4,294,967,297=641\times 6,700,417$. As of 2025, only five Fermat primes are known: $F_0=3$, $F_1=5$, $F_2=17$, $F_3=257$, and $F_4=65537$.

Generalized Fermat numbers extend this concept to arbitrary even bases. The most common definition used in distributed computing is numbers of the form

$F_{b,n}=b^{2^n}+1$

where $b>1$ is an even integer and $n\ge 1$. (If $b$ is odd, $b^{2^n}+1$ is always even and thus composite for $b>1$.) The PRIVATE GFN SERVER uses the broader definition allowing two coprime bases $a$ and $b$:

$a^{2^n}+b^{2^n},\gcd (a,b)=1,a>b>0.$

A prime of this form is called a Generalized Fermat prime (GFP). For a GFN to be prime, the exponent must be a power of 2; otherwise the number has a non-trivial algebraic factor.

Because of the relative ease of proving their primality and the abundance of candidates, many of the largest known prime numbers are generalized Fermat primes. The search for GFN primes at small $n$ (GFN-8 through GFN-14) has been chiefly coordinated by PRIVATE GFN SERVER, while PrimeGrid handles the larger $n$ (GFN-15 and above).

Why PRIVATE GFN SERVER?

This is a PRIVATE SERVER to coordinate distribution of some mathematical work on Generalized Fermat numbers (GFN). The project fills a gap left by PrimeGrid: while PrimeGrid focuses on GFN-15 and higher (primes with over one million digits), PRIVATE GFN SERVER has systematically searched the smaller exponent ranges, particularly GFN-11 through GFN-16, including the exhaustive search of all bases $b<2,000,000,000$ for $n\le 14$.

Goal

Pierre de Fermat (1601–1665), after whom Fermat numbers are named. Generalized Fermat numbers extend his original sequence $F_n=2^{2^n}+1$ to arbitrary bases.

The server is open for everybody, but some BOINC experience is recommended because there is no help, no support, no forum and no badges. Project information, help, and discussion are found in the PrimeGrid GFN Number Crunching sub-forum.

To join, volunteers must create an account directly on the project website (account creation via the BOINC Manager or BAM! is not supported). An invitation code — PrimeGrid — may be required at registration.

An encrypted HTTPS connection is also available at https://boincvm.proxyma.ru:30443/test4vm/.

Project team / Sponsors

The project is operated by a volunteer known as stream, with assistance from the PrimeGrid community. The project has no institutional sponsor and receives no formal funding; it is maintained on a volunteer basis. The server is hosted at boincvm.proxyma.ru.

Applications and subprojects

PRIVATE GFN SERVER runs multiple subprojects, each targeting a different mathematical problem or algorithm:

Active

The active subprojects as of 2025 include:

• GFN-12 / GFN-13 / GFN-14 — primality testing of numbers $b^{2^n}+1$ for $n\in \{12,13,14\}$ using the LLR2 application (CPU) or genefer20 (GPU).
• LLR2 / PRST testing — validation and double-checking of LLR2/PRST primality tests across all LLR2/PRST subprojects. The PRST application, a successor to LLR2 developed for factorial and primorial candidates, also supports AVX-512 instructions.
• Factorial / Primorial Sieve — GPU-accelerated sieving for factorial and primorial prime candidates, supporting AMD, Intel, and Nvidia GPUs.
• Correction of PrimeGrid PPSE Sieve (started May 2025) — a new GPU-only project to double-check and extend PrimeGrid's PPSE sieving in the 2M–3M range for both Proth and Riesel candidates, expected to run for approximately 3–4 months.

Project statistics links (active)

• GFN-12/13/14 primes per project
• LLR2/PRST tests (all LLR2/PRST projects)
• Factorial / Primorial Sieve

Archived subprojects

The following sub-searches have been completed:

• GFN-11 First MEGA prime search
• GFN-12 First MEGA prime search
• GFN-13 First MEGA prime search
• GFN-14 First MEGA prime search
• GFN-15 First MEGA prime search
• GFN-16 First MEGA prime search
• GFN-12 Prime Search — Final statistic (0–2000M)
• GFN-13 Prime Search — Final statistic (0–400M)
• GFN-13 Consecutive Primes Hunt
• GFN-14 Prime Search — Final statistic (0–400M)
• GFN-14 Consecutive Primes Hunt Part 1 (22M–65M)
• GFN-14 Consecutive Primes Hunt Part 2 (65M–400M)

A landmark result: the search of all bases $b<2,000,000,000$ for exponents $n\le 14$ is now complete thanks to the PRIVATE GFN SERVER. Full statistics are available at Generalized Fermat Numbers and historical data at GFN Prime Search Status and History on PrimeGrid.

Software and computing

The BOINC platform logo. PRIVATE GFN SERVER uses BOINC to distribute mathematical work to volunteers worldwide.

The project uses several applications:

• genefer20 — an OpenCL GPU application created by Yves Gallot in 2020, performing fast probable primality tests for numbers of the form $b^{2^n}+1$ with the Fermat test. It implements the Efficient Modular Exponentiation Proof Scheme (Darren Li) and validates results with Gerbicz–Li error checking.
• LLR2 — a CPU application using the Lucas–Lehmer–Riesel primality test, with AVX-512 support.
• PRST — a beta successor to LLR2, intended for primorial and factorial prime searches; also AVX-512-capable. The Gerbicz check is a key advantage of PRST over the older PFGW application for detecting hardware errors.
• fpsieve — GPU sieve application for factorial/primorial candidates.

Work units in the LLR2 testing subproject are validated instantly (minimum quorum of 1). The Factorial/Primorial sieve subproject supports AMD, Intel, and Nvidia GPUs.

Relationship with PrimeGrid

PRIVATE GFN SERVER was conceived as a complement to PrimeGrid, sharing applications and mathematical goals with PrimeGrid's GFN subprojects. The project description explicitly directs participants to PrimeGrid's Number Crunching sub-forum for instructions, help, and background information.

Where PrimeGrid focuses on very large GFN primes ($n\ge 15$, producing primes with millions of digits), PRIVATE GFN SERVER concentrated on the smaller and more numerous candidates at $n\le 14$, enabling an exhaustive search that PrimeGrid's structure was not suited for. The Primorial Sieve completed on the GFN Server directly enabled PrimeGrid to begin its downstream Primorial Prime Search project.

Scientific results

The project's primary scientific contribution is the systematic documentation of the distribution of GFN primes, extending tables originally published by Dubner and Gallot. The completed searches for $n\le 14$ provide definitive data for number theorists studying the density and distribution of generalized Fermat primes.

Relevant scientific literature

• H. Dubner, W. Keller: "Factors of generalized Fermat numbers", Math. Comp. 64 (1995), 397–405.
• A. Björn, H. Riesel: "Factors of generalized Fermat numbers", Math. Comp. 67 (1998), 441–446.
• H. Dubner, Y. Gallot: "Distribution of generalized Fermat prime numbers", Math. Comp. 71 (2002), 825–832.
• Darren Li: "Efficient Modular Exponentiation Proof Scheme", arXiv:2209.15623 (2022) — the proof scheme implemented in genefer20 and used to validate PRIVATE GFN SERVER results.

Participation

To participate:

1. Download and install the BOINC client.
2. Visit the project website and create an account (the invitation code PrimeGrid may be required).
3. In the BOINC Manager, add the project URL: http://boincvm.proxyma.ru:30080/test4vm/

The project does not support account creation through the BOINC Manager directly, nor through BAM!. An HTTPS mirror is accessible at https://boincvm.proxyma.ru:30443/test4vm/.

Statistics for the project are tracked by BOINCstats, Free-DC, and other BOINC statistics aggregators. The Scottish Boinc Team (TSBT) is among the historically leading teams at this project.

External links

• PRIVATE GFN SERVER (official project URL)
• Wayback Machine archive of project pages
• genefer20 source code (GitHub)
• Generalized Fermat Numbers statistics
• GFN Prime Search Status and History (PrimeGrid)
• BOINCstats — project statistics