We now know that the number 42 can be written as the sum of three cubes:
\[ 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 \]
This computational breakthrough was achieved in a collaboration between Andrew Sutherland (MIT) and Andrew Booker (Bristol). They announced the result by both replacing their homepages with the expression – with the page title Life, the Universe and Everything.
Every cube of a whole number is within one of a multiple of nine, which means that a sum of three cubes must be within three of a multiple of nine. So numbers of the form $9k+4$ or $9k+5$ cannot be written as the sum of three cubes.
In 1992, Roger Heath-Brown conjectured that every other whole number can be written as the sum of three cubes, in infinitely many different ways. Mathematicians on the whole seem to have been convinced by Heath-Brown’s argument that this ought to be true – but actually finding ways to write any particular number as a sum of three cubes remains a difficult problem.
Noam Elkies in 2000 came up with a practical algorithm for finding these representations. Elkies, and other people using essentially the same technique, managed to find solutions for a lot of smallish numbers.
In 2015, Tim Browning recorded a video with Numberphile in which he explained the problem. At that time, the only numbers less than 100 for which the problem was unsolved were 33, 42 and 74. This video brought the problem to the attention of a wider audience, which led to a series of breakthroughs.
Inspired by the Numberphile video, within a few months Sander Huisman found a solution for 74:
\[ 74 = (−284650292555885)^3 + 66229832190556^3 + 283450105697727^3 \]
Numberphile recorded another video about Huisman’s solution, again with Tim Browning. Another mathematician, Andrew Booker, saw the video and decided to have a crack at the problem. He came up with a new algorithm, which is more efficient at finding a solution for a particular number like 33. (Elkies’ algorithm conducts a trawl for triples of cubes whose sum is a small number, and it’s a less efficient way to search for a solution to a particular instance of the problem.) At five past nine in the morning on the 27th of February 2019, Booker’s computer found a solution:
\[ 33 = 8866128975287528^3 + (-8778405442862239)^3 + (-2736111468807040)^3 \]
When Tim Browning heard the news, he excitedly replaced his homepage with the solution, which was how it initally became publicly known. (This caused some initial confusion: because he didn’t include any more details, some people assumed that Tim Browning himself had made the discovery; but he quickly set the record straight.)
Earlier today, Keith Conrad noticed that Andrew Sutherland’s homepage had been replaced by the expression for 42, and then found that Andrew Booker’s homepage was the same. In a comment on Math Overflow, Conrad reports that he wrote to Sutherland, who explained that he and Booker worked together on 42, “and the change of their websites to show only the representation of 42 was inspired by how Browning made Booker’s result for 33 known”.
That means the only remaining unsolved cases up to a thousand are 114, 165, 390, 579, 627, 633, 732, 906, 921, and 975.
Breaking news: Numberphile have just released an interview with Andrew Booker where he talks about the new result on 42, and future plans.
Sums of three cubes on Wikipedia.
The Density of Zeros of Forms for which Weak Approximation Fails by D.R. Heath-Brown.
Rational points near curves and small nonzero $|x^3 − y^2|$ via lattice reduction by Noam Elkies.