modu 0.1.1

Modular arithmetic in Python

Modu is a mathematical module for doing modular arithmetic in Python.

In a nutshell, Modu allows you defining sets of residue classes for given moduli. These are displayed with usual notations of modular arithmetic like n ≡ {0, ±2, +3} (mod 6) or, in Jupyter Notebooks, with instantaneous LaTeX formula rendering. New sets can be computed from existing ones using set operations (union, intersection, complement), as well as arithmetic operations (addition, negation, multiplication, division, exponentiation). Samples of integers can be obtained, displayed as tables (aligning same residues in same columns) and possibly transforming the elements by user-defined functions. Several fundamental results of number theory can be easily illustrated, like the sieve of Eratosthenes, Fermat's little theorem, Chinese remainder theorem and Dirichlet's theorem on arithmetic progressions.

Thanks to Python's operator overloading, Modu allows expressing advanced operations in very few keystrokes, often as one-liners. Here are few examples.

• integers having residue 0 modulo 2 (the even numbers):

>>> from modu import O
>>> O % 2
0 (mod 2)

• integers multiples of 2 or 3, defined by set union:

>>> O%2 | O%3
{0, ±2, +3} (mod 6)

• integers not multiples of 2 or 3, defined by complement of the previous set:

>>> ~(O%2 | O%3)
±1 (mod 6)

• testing whether 15 and 23 belong to the previous set (note: any prime number greater or equal to 5 belongs to this set!):

>>> 15 in ~(O%2 | O%3)
False
>>> 23 in ~(O%2 | O%3)
True

• integers multiples of 2 and 3 altogether, defined by set intersection--see Chinese remainder theorem:

>>> O%2 & O%3
0 (mod 6)

• checking whether two sets are equivalent or not:

>>> O%2 | O%3 == (0, 2, 3, 4) + O%6
True
>>> O%2 & O%3 == O%6
True
>>> O%2 | O%3 != O%6
True

Modu provides much more functions, like changing the display format or extracting / transforming samples using a table layout. Modu is best used as an interactive calculator in Python terminal sessions (REPL) or in Jupyter Notebooks with instantaneous LaTeX rendering. The prime target domain is research and education in number theory.

All those features are demonstrated in an interactive tutorial, the "Modutorial" Jupyter Notebook (modutorial.ipynb). This can be opened directly online by clicking here (binder): (be patient: you have to wait a bit...)

Modu is an open-source module distributed under the MIT license.