Finding boundaries for a Diophantine equation

Question

I have the following equation:

$$k(k+1)(kx+376-x)=376n(nx+2-x)$$

Where $x\in\mathbb{N}$, $x\ge3$, $k\in\mathbb{N}$, $k\ge3$, $n\in\mathbb{N}$ and $n\ge4$.

Now, when I want to look for integer solutions I have two questions:

1. When I use the range $3\le k\le10000$, in what range should I look for $n$ in order to guarantee that $x$ satiesfies $x\in\mathbb{N}$ and $x\ge3$?
2. When I use the range $3\le n\le10000$, in what range should I look for $k$ in order to guarantee that $x$ satiesfies $x\in\mathbb{N}$ and $x\ge3$?

Answer 1

$k(k+1)(kx+376-x)=376n(nx+2-x)\implies$

$\Bigl((94 - k + k^3) x + 188 (k - 1 + k^2)\Bigr)^2 - 94 (94 - k + k^3) \Bigl(2 - x + 2 n x\Bigr)^2 = 376 k (k^2 - 1) (187 + 94 k)$

For $k\geq-4$ this Pell equation.

For k=-4..100 and without conditions for $x$ and $n$ equation has only solutions (k,x,n):

(7,47,2)
(9,1081,-1)
(23,-235,6)
(31,-141,-8)
(33,-47,9)
(45,-940,-15)
(47,-19,-12)
(47,-572,-16)
(47,8,24)
(47,10,-22)
(47,74,18)
(48,3,33)
(48,193,18)
(56,-658,-21)
(64,846,-26)
(87,141,-42)
(93,376,47)
(93,-128,-45)
(95,9,-57)
(95,128,-48)
(95,-376,48)