Hi, I would like to implement a Diophantine equations module for Sympy.
You can find my pull request here <https://github.com/sympy/sympy/pull/2024>.
It's still not merged.
I hope to solve following classical Diophantine equations. All the
variables and constants
used here are integers.
1)* **a1x1 + a2x2 + a3x3 + ...+ anxn = b* (Linear diophantine equation)
Here *a1, **a2, ... **an* and b are constants.If solvable (there is a
condition to determine this),
solving this equation means expressing any two variables using other
variables and an
arbitrary integer* *n. i.e. solution is given by
x1 = x1, x2 = x2, ... xn-2 = xn-2 , xn-1 = f( x1, x2, ... xn-2, n), xn-1 =
g( x1, x2, ... xn-2, n)
f and g are functions to be determined.
2) x12 + x22 + x32 + ... xn2 = k
Here k is a non-negative constant. There will be a number of solutions
depending on
n and k. Solving this means assigning constants *a1, **a2, ... **an* to x*i
*'s respectively.
3) x12 + x22 + x32 + ... xn2 = xn+12 (extension of Pythogorean equation)
Solving this is pretty standard. There is a general primitive solution set
using n relatively
prime integers. All other solutions can be obtained by multiplying those
equations by
an arbitrary integer.
4) x2 + axy + y2 = z2
Here a is a constant. If z is a variable, a general solution can be given
to this equation
using a and three arbitrary integers. If z is a constant actual solutions
can be given.
5) x2 - Dy2 = m2 (Pell's equation)
Here D and m are constants. This has either no solution or infinitely many
solutions.
ax2 - by2 = 1 and ax2 + bxy + cy2 + dx + ey + f = 0 can also be solved with the
light
of Pell's equation.
Lot of Diophantine equations can be converted to one of these forms. Addition
of this kind
a module will be a huge enhancement for Sympy. I would like to know how I
can improve
this. Thanks in advance.
References
[1] An Introduction to Diophantine Equations*, Andreescu*, Titu, *Andrica*,
Dorin, *Cucurezeanu*, Ion
[2] http://mathworld.wolfram.com/DiophantineEquation.html
[3] http://en.wikipedia.org/wiki/Diophantine_equation
Regards,
Thilina Rathnayake.
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