In this post, we are interested in finding all integer and rational solutions for a general binary quadratic equation:
ax2+bxy+cy2+dx+ey+f=0,\labeleq:general
for any integer coefficients a,b,c,d,e,f. The equation might degenerate; we only point it out when it does but do not solve it, as it’s typically easier to solve a degenerate equation.
This post mainly serves as an entry point for a series of posts and also contains a little bit of the story behind the posts. That is why this post has index 0 in the title.