Calculation Results for the Square Root of 2 up to 1,200,000,000,100 Digits

The Square Root of 2

Calculation of the first 1,200,000,000,100 digits of the square root of 2, both in decimal and hexadecimal digits.

\[\sqrt{2} = 2^{1/2} = 1.4142135623\ldots\]

The square root of 2 is the positive number that gives 2 when multiplied by itself, and it is the length of the diagonal of a unit square by the Pythagorean theorem. It was proved irrational in antiquity, and the proof is simple enough that it comes out in secondary school mathematics. For more information on the square root of 2, see this Wolfram Mathworld entry.

Newton’s Method (\(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)) was used for computation, as it simply finds a closer approximation per step and is the cheapest of all the constants y-cruncher can compute. Note that this is not a world record, as the record for the square root of 2 stands at 28,000,000,000,000 digits as of June 2025. This computation was mainly done to test a new system.

This computation of 1,200,000,000,100 decimal digits and 996,578,428,550 hexadecimal digits by Seungmin Kim was done from Fri Aug 15 15:15:55 2025 to Sat Aug 16 10:51:21 2025.
Validation file generated by y-cruncher v0.8.6 Build 9545-icx:
https://archive.org/download/squareroot2_1_2T/Sqrt%282%29%20-%2020250816-105139.txt

As with all other mathematical constants, I used y-cruncher by Mr. Alexander J. Yee for this computation. This program is commonly used for stress testing and benchmarking overclocked PC builds (obviously this program performs a very rigorous computation), along with the fellow mathematical computing program Prime95 and the linear algebra program Linpack.

Computation:
System information:
Operating System: Linux 5.15.134+release+2.10.0r8-amd64 x86_64
Processor(s):
Name: AMD EPYC 7742 64-Core Processor
Logical Cores: 256
Physical Cores: 128
Sockets: 2
NUMA Nodes: 2
Base Frequency: 2,249,960,768 Hz

I have used two AMD EPYC Rome sockets that are compatible with AVX2 SIMD vector operations that are used crucially in vector-heavy programs like y-cruncher. Although 0.98 TiB of usable memory was available, only 64.0 GiB was allocated to the computation, so almost all of the intermediate data went to swap storage instead of RAM. The number of cores therefore did not assist my operation well, and the multi-core efficiency stayed very low even though the CPUs are far more capable than the ones used for my earlier computations.

Start Date: Fri Aug 15 15:15:55 2025
End Date: Sat Aug 16 10:51:21 2025
Total Computation Time: 43284.521 seconds
Start-to-End Wall Time: 70525.466 seconds
CPU Utilization: 703.05 % + 164.70 % kernel overhead
Multi-core Efficiency: 2.75 % + 0.64 % kernel overhead

Memory:
Working Memory: 67,483,658,368 ( 62.8 GiB)
Total Memory: 68,719,476,736 ( 64.0 GiB)
Logical Largest Checkpoint: 2,773,058,577,120 (2.52 TiB)
Logical Peak Disk Usage: 4,714,404,148,888 (4.29 TiB)
Logical Disk Bytes Read: 50,517,146,935,148 (45.9 TiB)
Logical Disk Bytes Written: 39,169,298,487,668 (35.6 TiB)

Disk operation was a tiny fraction of what the harder constants require, which is why twice the digits of my Apéry’s constant computation finished in under a day instead of a month. Newton’s Method is the cheapest algorithm here, and the base conversion from binary to decimal took longer than the square root itself. No ECC corrected errors occurred during the computation.

To view the digits for the square root of 2, you can download them from This Link (note that the total size is over 1 TB, but the link will simply redirect to a registry with a download link).

Archive (Just a Registry):

https://archive.org/details/squareroot2_1_2T

Decimal (Data Warning):

https://archive.org/download/squareroot2_1_2T/Sqrt%282%29%20-%20Dec%20-%20Newton.zip

Hexadecimal (Data Warning):

https://archive.org/download/squareroot2_1_2T/Sqrt%282%29%20-%20Hex%20-%20Newton.zip

Validation:

https://archive.org/download/squareroot2_1_2T/Sqrt%282%29%20-%2020250816-105139.txt

Note that the digits are released under an Attribution-NonCommercial-NoDerivatives 4.0 International License, which prohibits commercial use and distribution of remixed, transformed, or built-upon versions without consent. Proper attribution and indication of changes are required even if it is not a prohibited use case.

Special thanks to Mr. Alexander J. Yee for developing and releasing y-cruncher, and to the Internet Archive for hosting the computed digits.