How to solve quadratic Diophantine equation with 3 variables

Question

Given the equation:

$3x^2 - x - 3y^2 + y = 3n^2 - n$

I'd imagine solving this involves techniques for solving Diophantines? Or am I wrong?

Could someone point me in the right direction?

Answer 1

Too long for a comment:

• Renaming y and n with a and b, we have $f(x)=f(a)+f(b),$ where $f(t)=3t^2-t.$

• It is obvious that if $(a,b,x)$ represents a solution, then so does $(b,a,x).$

• Since $f(0)=0,\quad(0,x,x)$ and $(x,0,x)$ always constitute solutions.

• Therefore, it is enough to take into consideration the case $a\le b$ with $ab\neq0.$

• This being said, for $-100\le a\le b\le100,$ we have the following non-trivial solutions :

$\qquad\qquad\qquad\qquad$

• I offer this numerical data in the hope that it will aid future analytic answers.