Fantastic, thanks! For your first point about needing $n$ of the first $\binom{n+1}{2}$ natural numbers: should that be $n$ of the first $n^2$, since the number of pairs of $n$ naturals is $n^2$ because repetition is allowed?

This has been asked before (math.stackexchange.com/questions/924931/…). I disagree with the answers listed below - the issue isn’t that you need to state that the sets aren’t empty. Rather, it’s that the statement proved isn’t the statement that needs to be proved.

@r.e.s. My apologies, yes, you're correct. I forgot about the cost of all the loops up to $k$ itself. Thanks for pointing that out!

If the largest number in the solution is $k$, then the code will take something on the order of $8k^3$ steps to terminate. For $n = 33$ the numbers needed are on the order of $10^{15}$ and you’ll need about $10^{45}$ steps to finish. The smallest solution for 33 uses numbers on the order of $10^{15}$, so the code will need to take about $10^{45}$ steps to finish.