One easy-to-state open problem in number theory is whether every integer not of the form is the sum of three cubes. Some solutions are easy to find, while others are notoriously difficult.
For example 29 can be written as , or even .
It is clear that cannot be written as the sum of three cubes since .
There also exist infinite families of solutions in certain cases (though these need not represent all solutions). For example,
For a long time it was not known whether 30 can be written as the sum of cubes. The following is the “simplest” way this can be done, found as recently as 1999.
It is still not known whether 33, 42 or 74 can be written as the sum of three cubes (edit: it is now known that 74 can be – see )! In these cases no solution has been found when any of the cubes is less than in magnitude .
One of the computational algorithms to find solutions to where is relatively small, proposed by Elkies, involves converting the equation to and then considering rational points close to the curve . This curve is covered by small parallelograms and the problem is converted to finding lattice points in a pyramid using basis reduction followed by the Fincke-Pohst algorithm .
 Hisanori Mishima, Chapter 4. n=x^3+y^3+z^3
 D.J. Bernstein, http://cr.yp.to/threecubes.html
 A.-S. Elsenhans and J. Jahnel, New sums of three cubes, Math. Comp. 78 (2009), 1227–1230, available here.
 Sander G. Huisman, Newer sums of three cubes, http://arxiv.org/abs/1604.07746, April 2016.