I wanted to find pair of (integers) numbers (actually just 3 of them) such that:
$$9U=9K_1 + 4 K_1^2$$
I've tried choosing a fixed integer starting at 1 for U and then had mathematica/wolfram find $K_1$. So far I've tried up to 6 with no luck. However, I wasn't sure if this even had a solution and my knowledge in number theory is quite limited. So I thought it would be wise to ask the expert community on this.
Just in case this is impossible to solve, I'd be sort of ok with numbers such that $U,K_1$ are at least $ \epsilon < 0.5$ close to an integer. This is not the ideal case but if its impossible I accept approximations or I need to change my conditions to make this problem solvable.
If you are curious this equation came from trying to find integers that satisfy:
$$ 9U = 3K_1 + 2 K_1 K_2 + 3 K_2 $$
for some nice properties of my problem choosing $4K_1 = 2K_2 \iff 2K_1 = K_2$. Though I guess we could change that condition if necessary but it wouldn't be ideal.
The solution set is $\{K_1=3a, U=3a+4a^2 \mid a\in \mathbb Z\}$