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175 billion digits of Gamma(1/5)

The Gamma function is a continuous extension of the factorial function. It is defined for real numbers and complex numbers with the following formula:

Γ(z)=∫0∞tz−1e−tdt,ℜ(z)>0\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} \ \mathrm{d}t, \quad \Re (z) > 0

It is currently unknown if Gamma(1/5) is irrational or transcendental. If it turned out to be rational, its decimal expansion would eventually fall into a repeating cycle. No such repetition has been found (much less proven), and many believe Gamma(1/5) to most likely be transcendental.

y-cruncher

Using y-cruncher, I calculated 175 billion digits of Gamma(1/5) with this laptop in my dorm (window open for ventilation):

laptop_in_window.webp

Component Specification
Processor AMD Ryzen 5 7640U
Topology 12 threads / 6 cores / 1 socket
Usable Memory 62.0 GiB
Usable Storage 1.14 TiB

I ran the calculation twice to verify my results:

The first calculation used the Vidunas-Lamprecht (2024) algorithm provided by y-cruncher and was run on a framework laptop running NixOS.

Γ(15)=5(5+5)(−1/10)⋅π(−1/5)⋅2(−19/50)⋅H1(−2/5)⋅H2(−1/5)\Gamma(\tfrac{1}{5}) = \frac{ \sqrt{5} }{ (\sqrt{5} + 5)^{(-1/10)} \cdot \pi^{(-1/5)} \cdot 2^{(-19/50)} \cdot H_1^{(-2/5)} \cdot H_2^{(-1/5)} }

where

H1=∫01dz1−z5,H2=∫01zdz1−z5H_1 = \int_0^1 \frac{\mathrm{d}z}{\sqrt{1-z^5}}, \quad H_2 = \int_0^1 \frac{z \ \mathrm{d}z}{\sqrt{1-z^5}}

A second computation, this time performing the Incomplete Gamma Approximation algorithm, matched the results of the first computation. Here are some interesting stats from these computations:

Method Read Written Computation Time
VL 503 TiB 443 TiB 15 days, 22 hours
IGA 356 TiB 316 TiB 52 days, 7 hours

My understanding is that the first method (Vidunas-Lamprecht) is intrinsically more efficient than the Incomplete Gamma Approximation; however, most of the observed performance gap is due to hardware differences. The laptop had an NVMe SSD and a much newer CPU compared to the older hard drives and processor in the Debian server used for verification.

Over the two runs combined, more than 1.6 Pebibytes (PiB) of data (about 1.64 * 106 GiB) was read and written to disk. No I/O errors were detected, and all drives remain in full working condition.

Results

Here are the last digits I computed (numbered 174,999,999,901 to 175,000,000,000), broken into groups of 10 for clarity: 

6180267994 6874068253 4568233834 0586791383 1778803247
0727912102 7020053472 8225828190 3835497822 1028929467

Aftermath

When I started this computation, the record for Gamma(1/5) (computed by Dmitriy Grigoryev) was 135 billion decimal digits, announced on February 20, 2025; however, a new computation of 220 billion digits was completed on May 26, 2025, just 5 days before I completed my 175 billion digit computation.

However, this was still a fun personal challenge that provided interesting results on the computation time comparison between a modern laptop and an older server for I/O-bound calculations.

Last updated June 22, 2026