Integer solutions to the equation $y^2\pm3xy+x^2=\ell$

Question

I would like to know if the number of integer points on curves of the form $$y^2\pm3xy+x^2=\ell$$ where $\ell$ is a prime number, is finite or not. Also, is there a name for such equation ? Thank you !

EDIT : I require solutions $x,y$ to be positive, hence it is immediate that $y^2+3xy+x^2=\ell$ has finitely many solutions.

Answer 1

Assume either $x$ or $y$ is even. WLOG $x=2z$. So $y^2-6yz+4z^2=\ell$. This means $(y-3z)^2-5z^2=\ell$. Denote $y-3z=t$. We have $t^2-5z^2=\ell$. It is a generalized Pell equation. If it has a solution, it has infinitely many solutions, see this text by K. Conrad, Section 3. Or this text by, I believe, his brother. See also this question.

Answer 2

Every indefinite binary quadratic form, integer coefficients and discriminant positive but not a square, has an infinite automorphism group. In the special case of your $x^2 - 3 xy + y^2,$ this can be described with what the contests call "Vieta Jumping." Suppose you have a solution, with integers $x > y > 0,$ to $$ x^2 - 3xy + y^2 = \ell $$ You get an infinite sequence of solutions, increasing in the first entry, by repeatedly applying $$ (x,y) \mapsto (3x-y,x) $$ Note that $x>y>0$ tells us $x-y > 0,$ $2x-y > x,$ indeed $3x-y > 2x.$ So the first number keeps growing, and both stay positive.

For example, to represent the prime $5$ by $x^2 - 3xy + y^2,$ we get a sequence of solutions $$ (4,1) \mapsto (11,4) \mapsto (29,11) \mapsto (76,29) \mapsto (199,76) \mapsto (521,199) \mapsto \cdots $$

Answer 3

$y^2\pm3xy+x^2=\ell\implies (2 y \pm 3 x)^2 - 5 x^2 = 4 \ell$

This Pell equation with infinite set integer solutions for $\ell$=5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, ..., and no solutions for others $\ell$. But positive solutions are only for $y^2-3xy+x^2=\ell$.