$ k x^2 +4x = n $, Algorithm or any other method needed I want to find any $n < 10^{18} $ so that the equation below has at least two pairs of solutions $(k, x)$
$ k  x^2 +4 x  = n $
constraints: $x > 10^6; \; x > k ; \; k, x \in \mathbb{N}$
I can not solve this mathematically, nor I can think of any fast brute-force algorithm.
Is there any integer programming method which deals with such kind of problem?

Take $$(k_1,x_1)=(500004,1000002)$$ $$(k_2,x_2)=(500000,1000006)$$
Then, $$k_1x_1^2 + 4x_1 = k_2x_2^2 + 4x_2 = 500006000022000024$$

More generally, any integer $S$ can be used to produce an at-least-twice-expressible number: $$(k_1,x_1)=(S+4, 2S+2)$$ $$(k_2,x_2)=(S,2S+6)$$ both result in $$n=4(S+1)(S+2)(S+3)$$
