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

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Author: Thomas G. Close (

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)


Diophantine is available from the Python Package Index ( and can be installed with the command

pip install diophantine

Alternatively, the master branch can be installed from using the setuptools install command

python install

from a clone of the GitHub repository ( or simply adding the cloned directory to your PYTHONPATH.

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