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Consider the following example Diophantine equation

-118w + 989x + 918y  -512z == 0

You can solve this in sympy with:

import sympy
from sympy.solvers.diophantine import diophantine 
w, x, y, z = sympy.symbols('w x y z')
a, b, c, d = -118, 989, 918, -512
diof = list(diophantine(a*w + b*x +c*y +d*z ))[0]
print(diof)

which gives:

(t_0, 118*t_0 + 2*t_1, 1690468*t_0 + 28937*t_1 + 256*t_2, 3031184*t_0 + 51887*t_1 + 459*t_2)

Explanation: This solution allows you to pick three integers t_0, t_1, t_2 freely and from there you can compute the values of w, x, y, z.

This is fine except the coefficients are always much larger than they need to be. If you solve the same system in Mathematica you get:

x = 2 c_2, y = 9 c_2 + 256 c_3 + 81 w, z = 20 c_2 + 459 c_3 + 145 w, c_2 element Z and c_3 element Z

or:

enter image description here

Rewritten in the same style as the sympy solution this would be:

(t_0, 2*t_1, 9*t_1 + 256*t_2 + 81*t_0, 20*t_1 + 459*t_2 + 145*t_0)

Is it possible to get sympy to give solutions to linear Diophantine equations with similarly small coefficients?

Typos fixed in the sympy code.

Simd
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  • cf [this question](https://math.stackexchange.com/questions/3446084/how-can-we-find-the-minimal-solutions-to-a-linear-diophantine-equation) – smichr Oct 28 '21 at 22:46
  • sympy solution is showing 3 variables, t0, t1, and t2 whereas mathematica is only showing c2 and c3. so coefficient similarity is not comparable. – ferdy Oct 30 '21 at 00:13
  • @ferdy it's the same. I have updated the question to show this. – Simd Oct 30 '21 at 11:09
  • @Anush, right, however notice the output format of diophantine solution, x is based on t_0 and t1 while in mathematica x is based only on t_1. Looking at the [sympy diopantine docs](https://docs.sympy.org/latest/modules/solvers/diophantine.html?highlight=diophantine), there is no option for diophantine to give solution where the x (the second term) will be based on t_1 only. So the current diophantine cannot solve what mathematica solution format is doing. You may go to [sympy github repo](https://github.com/sympy/sympy) and ask a feature to possibly simulate mathematica solution format. – ferdy Oct 31 '21 at 02:10

0 Answers0