For a^3 + b^3 + c^3 = 42,

You can enumerate a,b pairs and then you need to check whether the "locked in" value of c^3 is a cube.

However imagine it takes 1ns to validate a given pair [a,b].

The eventual solution was [-80538738812075974, 80435758145817515, 12602123297335631].

Since no combination of 3 positive (and therefore small) numbers has worked, we know that one of a,b,c are negative. Let's assume at least one of a,b are negative since it doesn't matter how we allocate them.

To reach the final pair of a = 80435758145817515 (the smaller positive integer) and b = -80538738812075974, you have to increment "a" (starting from 0) 80435758145817515 times and decrement "b" (starting from 0) 80538738812075974 times.

That is 80538738812075974*80435758145817515 possible combinations.

Let's assume each one takes 1 ns (which I believe is fairly optimistic at least for a single machine)

That results in a runtime of 6.5e+24 seconds, aka 2.1e+17 years. No matter how many machines you add, the brute force approach does not appear to be feasible.

I am interested to learn more about how they solved it if not brute force.