Skip to main content
This is a pre-production deployment of Warehouse. Changes made here affect the production instance of PyPI (
Help us improve Python packaging - Donate today!

A python package for finding small solutions of systems of diophantine (integer algebra) equations

Project Description

Diophantine ( is a Python package for finding small (integer) solutions of systems of diophantine equations (see It is based on PHP code by Keith Matthews (see that implements the algorithm described in (see for a list of associated publications), which uses the LLL algorithm to calculate the Hermite-normal-form described in the paper:

Extended gcd and Hermite normal form algorithms via lattice basis reduction, G. Havas, B.S. Majewski, K.R. Matthews, Experimental Mathematics, Vol 7 (1998) 125-136

(please cite this paper if you use this code in a scientific publication)

There are two branches of this code in the GitHub repository (see, ‘master’, which uses the sympy library and therefore uses arbitrarily long integer representations, and ‘numpy’, which uses the numpy library, which is faster but can suffer from integer overflow errors despite using int64 representations

To find small solutions to a system of diophantine equations, A x = b, where A is a M x N matrix of coefficents, b is a M x 1 vector and x is the N x 1 vector, use the ‘solve’ method in the module, e.g.

>>> from sympy import Matrix
>>> from diophantine import solve
>>> A = Matrix([[1, 0, 0, 2], [0, 2, 3, 5], [2, 0, 3, 1], [-6, -1, 0, 2],
                [0, 1, 1, 1], [-1, 2, 0,1], [-1, -2, 1, 0]]).T
>>> b = Matrix([1, 1, 1, 1])
>>> solve(A, b)
[ 1],
[ 0],
[ 0],

The returned solution vector will tend to be one with the smallest norms. If multiple solutions with the same norm are found they will all be returned. If there are no solutions the empty list will be returned.

Diophantine is released under the MIT Licence (see Licence for details)

Author: Thomas G. Close (

Release History

Release History

This version
History Node


Download Files

Download Files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

File Name & Checksum SHA256 Checksum Help Version File Type Upload Date
Diophantine-0.1.tar.gz (6.6 kB) Copy SHA256 Checksum SHA256 Source Jul 29, 2016

Supported By

WebFaction WebFaction Technical Writing Elastic Elastic Search Pingdom Pingdom Monitoring Dyn Dyn DNS Sentry Sentry Error Logging CloudAMQP CloudAMQP RabbitMQ Heroku Heroku PaaS Kabu Creative Kabu Creative UX & Design Fastly Fastly CDN DigiCert DigiCert EV Certificate Rackspace Rackspace Cloud Servers DreamHost DreamHost Log Hosting