So I'm wondering if there's a way to capture the weirdness of these solutions (x, y, z) in Z^3 to x^3 + y^3 + z^3 = N, namely the extreme largeness of the "smallest" solutions for some integers N. I don't recall if anyone defined "smallest", probably because several notions of smallest were / are believed to coincide in these cases. But let's use (distance? No. Better is:) squared distance of (x, y, z) to the origin, D(x,y,z) = x^2 + y^2 + z^2. The goal: to measure the extent that (x,y,z) |—> x^3 + y^3 + z^3 takes large input to small output. How best to do this? The values of distance^2 (henceforth: dist2) from the origin for the input has one distribution while those values for the output has another. [Here we switch to using just x or y for an entire input point.] Each datum in a distribution has some fraction of all data less than it and some fraction greater than it. In this case these fractions will add up to 1. To find how much a function "disorders" its input over a finite measure domain, we can calculate the average value E(fr_input(x) - fr_output(x)) of the difference between input fraction (the measure of the set of points fr_input(x) = measure of {y in domain | dist2(y) <= dist2(x)}, and the output fraction fr_output(x) = measure of {y in domain | f(y) <= f(x)} where these two numbers are each between 0 and 1. Hence their difference and its expected or average value will also lie in the unit interval. (But I suspect the maximum amount you can disorder something to be less than !. Even if you turn something upside down it's only disordered on average by 1/2.) —Dan James Buddenhagen writes: ----- ... https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf Also, D.R. Heath-Brown has conjectured that if k is not +-4 mod 9 then there are infinitely many solutions in integers to x^3 + y^3 + z^3 = k. Apparently this conjecture is thought to be true by Booker and (some?) others familiar with the problem. -----