In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for an integer to equal such a sum is that cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient.
Variations of the problem include sums of non-negative cubes and sums of rational cubes. All integers have a representation as a sum of rational cubes, but it is unknown whether the sums of non-negative cubes form a set with non-zero natural density.
Small cases
A nontrivial representation of 0 as a sum of three cubes would give a counterexample to Fermat's Last Theorem for the exponent three, as one of the three cubes would have the opposite sign as the other two and its negation would equal the sum of the other two. Therefore, by Leonhard Euler's proof of that case of Fermat's last theorem, there are only the trivial solutions
For representations of 1 and 2, there are infinite families of solutions
(discovered by K. Mahler in 1936) and (discovered by A.S. Verebrusov in 1908, quoted by L.J. Mordell). These can be scaled to obtain representations for any cube or any number that is twice a cube. There are also other known representations of 2 that are not given by these infinite families:
However, 1 and 2 are the only numbers with representations that can be parameterized by quartic polynomials as above. Even in the case of representations of 3, Louis J. Mordell wrote in 1953 "I do not know anything" more than its small solutions
and the fact that each of the three cubed numbers must be equal modulo 9.
Computational results
Since 1955, and starting with the instigation of Mordell, many authors have implemented computational searches for these representations. used a method of involving lattice reduction to search for all solutions to the Diophantine equation
for positive at most 1000 and for , leaving only 33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and 975 as open problems in 2009 for , and 192, 375, and 600 remain with no primitive solutions (i.e. ). After Timothy Browning covered the problem on Numberphile in 2016, extended these searches to solving the case of 74, with solution
Through these searches, it was discovered that all that are unequal to 4 or 5 modulo 9 have a solution, with at most two exceptions, 33 and 42.
However, in 2019, Andrew Booker settled the case by discovering that
In order to achieve this, Booker exploited an alternative search strategy with running time proportional to rather than to their maximum, an approach originally suggested by Heath-Brown et al. He also found that
and established that there are no solutions for or any of the other unresolved with .
Shortly thereafter, in September 2019, Booker and Andrew Sutherland finally settled the case, using 1.3 million hours of computing on the Charity Engine global grid to discover that
as well as solutions for several other previously unknown cases including and for .
Booker and Sutherland also found a third representation of 3 using a further 4 million compute-hours on Charity Engine:
This discovery settled a 65-year old question of Louis J. Mordell that has stimulated much of the research on this problem.
While presenting the third representation of 3 during his appearance in a video on the Youtube channel Numberphile, Booker also presented a representation for 906:
The only remaining unsolved cases up to 1,000 are the seven numbers 114, 390, 627, 633, 732, 921, and 975, and there are no known primitive solutions (i.e. ) for 192, 375, and 600.
Popular interest
The sums of three cubes problem has been popularized in recent years by Brady Haran, creator of the YouTube channel Numberphile, beginning with the 2015 video "The Uncracked Problem with 33" featuring an interview with Timothy Browning. This was followed six months later by the video "74 is Cracked" with Browning, discussing Huisman's 2016 discovery of a solution for 74. In 2019, Numberphile published three related videos, "42 is the new 33", "The mystery of 42 is solved", and "3 as the sum of 3 cubes", to commemorate the discovery of solutions for 33, 42, and the new solution for 3.
