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?

  • $\begingroup$ This becomes elaborate fairly quickly. Why do you want to know? Anyway, multiply through by $12$ and complete the squares. $\endgroup$ – Will Jagy Jan 3 '16 at 4:02
  • $\begingroup$ @WillJagy I need to solve this equation as a piece in a computational problem. $\endgroup$ – Thevesh Theva Jan 3 '16 at 4:03
  • $\begingroup$ math.stackexchange.com/questions/794510/… For this equation, one can dispense with the equation of Pell. But why are you asking questions? Still the answers don't interest you. $\endgroup$ – individ Jan 3 '16 at 4:24
  • $\begingroup$ @individ thank you very much! What do you mean by the answers not interesting me? $\endgroup$ – Thevesh Theva Jan 3 '16 at 7:15

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 :


  • I offer this numerical data in the hope that it will aid future analytic answers.
  • $\begingroup$ @individ: You can turn that into an answer, and check to see if by any chance there aren't other families of solutions, by comparing it to the above-given list. $\endgroup$ – Lucian Jan 3 '16 at 8:02
  • $\begingroup$ The bug was. Already fixed. Well and what sense was to write all these numbers? That complex equation? $$3n^2-n+3y^2-y=3x^2-x$$ Had written such solution. $$n=18p^2+18ps+p+\frac{s(9s+1)}{2}$$ $$y=24p^2+24ps-2p+s(6s-1)$$ $$x=30p^2+30ps-p+\frac{s(15s-1)}{2}$$ $\endgroup$ – individ Jan 3 '16 at 8:03
  • $\begingroup$ A more simple option. $$n=(6k+7)t$$ $$y=(18k^2+42k+24)t-k-1$$ $$x=(18k^2+42k+25)t-k-1$$ $\endgroup$ – individ Jan 3 '16 at 8:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.