Pascal Mersenne Rows

This is one of the approaches I am exploring for representing numbers using Pascal structures. Here I write certain powers of two as Pascal rows in Mersenne bases. The proof is in the mathematical note; Python and C programs are included to reproduce examples. Türkçe

Result and example

The note proves when the chosen base and row give a digit representation without carrying.

For example, these base-127 digits represent 2^63:

1 9 36 84 126 126 84 36 9 1

Here 126 is a single digit. Every digit is below 127, so no carrying is needed. This result is not a general primality test.

How I did it

I used the Pascal coefficients in the binomial theorem as digits in the chosen base. The programs compute those coefficients, check that each fits the base, and verify that their value equals the expected power of two.

Formulas, proofs, and references are in the mathematical note (text version). The figure explains the positions in the row.

How to run

Python 3 is sufficient; no third-party packages are needed. From the repository root:

python3 mersenne_rows.py 7 full

On Windows use py -3 mersenne_rows.py 7 full. Expect clean=yes, value(mod)=ok, and value(full)=ok. The reported 64 bits is the length of this example's result, not a limit on the program's number size.

Setup and tests

Installation is optional. To copy the package into your home folder:

python3 setup/install.py "$HOME/Mersenne"

Windows: setup\install.bat "%USERPROFILE%\Mersenne". macOS: ./setup/install.command "$HOME/Mersenne". The installer does not download Python or GMP and does not overwrite conflicting package files automatically. Run tests with python3 -B tests/test_local.py.

The optional C version needs a C compiler and GMP:

cc -O2 -o mersenne_rows_gmp mersenne_rows_gmp.c -lgmp
./mersenne_rows_gmp 7 full

Files and terms

• setup/: installation; tests/: program and installation checks.
• tables/: tables; coordinates_en.pdf: figure. Turkish versions are under tr/.
• note.md, note.pdf: text and PDF versions of the mathematical note.
• Licenses: all source code, launchers, setup files and tests are covered by MIT. English and Turkish documentation, PDFs, figures, tables and citation metadata are covered by CC BY 4.0. Copyright © 2026 James Augustus. These licenses apply to their respective materials, rather than offering a choice of license for every file.
• CITATION.cff: citation information.

James Augustus · jamesaugustus1313@gmail.com