About

This repository is mainly dedicated to solving the following problem: for $N \in \mathbb{N}$ find a natural solution, i.e. $a, b, c \in \mathbb{N}$, such that the following equation holds:

$$\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} = N.$$

This problem is equivalent to finding rational points with the same sign on the cubic, defined by this equation:

$$x^3 + y^3 + z^3 + (1 - N) (x^2 y + x^2 z + y^2 x + y^2 z + z^2 x + z^2 y) + (3 - 2 N) x y z = 0$$

To do that, we perform a projective transformation (with integer coefficients), which maps this cubic onto cubic with equation:

$$y^2 z = x^3 + a x z^2 + b z^3$$

also known as Weierstrass form.

This form is very useful when it comes to addition of the points on the cubic curve. So, now we're looking for any rational point on this cubic and once we find it, we add it to itself and look at the inverse image of the point. This inverse image has a rational coordinates and if all signs are the same -- we win and the problem is solved. If not, we continue adding point to itself. There is a chance, however, that this point is from finite cyclic subgroup of group of all rational points on our Weierstrass cubic. If this happens, we have to search for point from another subgroup and start process from the top.

Requirements

First you need to install python3 (I think with any version of python3 you will be good to go). And you will also have to install the following packages for python3, which you can install via pip:

• matplotlib
• sympy
• numpy

Usage

To find the solution for the problem run Programs/main/main.py. You can also run the file Programs/main/weierstrass_form/weierstrass_main.py to find a Weierstrass form for the given cubic (if possible).