# Status of the Diophantine module

Hi All,

In my project proposal for a Diophantine equation for SymPy, I mentioned the following five deliverables.

1. Linear Diophantine equation $a_1x_1 + a_2x_2 + . . . + a_nx_n = b$:
I implemented solutions for linear diophantine equations, you can access this functionality through `diop_linear()`.

2. Simplified Pell equation, $x^2 - Dy^2 = 1$:
Not only I implemented solutions for simplified Pell equation, I completely solved the general binary quadratic equation $ax^2 + bxy + cy^2 + dx + ey + f = 0$.

3. The equation, $x^2 + axy + y^2 = z^2$:
I implemented solutions for more general ternary quadratic equation $ax^2 + by^2 + cz^2 + dxy + eyz + fxz = 0$.

4. Extended Pythagorean equation,  $x_1^2 + x_2^2 + . . . + x_n^2 = x_{n+1}^2$:
I implemented solutions for slightly more general equation $a_1^2x_1^2 + a_2^2x_2^2 + . . . + a_n^2x_n^2 = a_{n+1}^2x_{n+1}^2$.

5. General sum of squares, $x_1^2 + x_2^2 + . . . + x_k^2 = n$:
This is a computationally hard problem and method I implemented finds only one solution. It’s quick and work for large $n$ but not complete. I also implemented a brute force version which finds all the solutions but it doesn’t work for larger $n$.