A seventy-year-old question with a finite answer.

Which whole numbers are a sum of three cubes? Negative integers count, so the cubes can cancel each other, and the numbers involved can get enormous. In 1953 Louis Mordell asked whether 3 had any representation besides 1³ + 1³ + 1³ and 4³ + 4³ + (−5)³. The first computer search ran on Cambridge’s EDSAC in 1954.

Part of the answer is easy. Every cube leaves 0, 1 or 8 after dividing by 9, so a number that leaves 4 or 5 can never be a sum of three cubes. For every other number, the conjecture of Roger Heath-Brown (1992) says there should be infinitely many representations. Nobody can prove it, and for the stubborn cases nobody had found even one.

Then came 2019. Andrew Booker found 33. Booker and Andrew Sutherland, with the volunteer grid Charity Engine, found 42 and a third representation of 3. 165, 579, 795 and 906 followed. Below 1000, seven numbers are left: 114, 390, 627, 633, 732, 921 and 975.

Why this one, of all the open problems

Most famous open problems are statements about all numbers at once. Proving them takes ideas nobody has had yet, and checking a claimed proof takes experts months. This one is different: a single line of three integers settles 114 for good, and a computer can check that line in microseconds. That is why the program here can pay a solver without a committee.

Why it is still hard

For 114, every solution whose smallest number (ignoring sign) is at most 10¹⁷ has been ruled out (Booker and Sutherland, 2019). A partial search in 2020 reached 10¹⁹ along part of the range. Epoch AI’s heuristic estimate puts the median size of the first solution around 4 × 10²³, millions of times past the complete search. Finding one more solution at the expected rate takes something like ten million times more search than the last one did. More hardware alone is unlikely to get there; it probably needs a new idea.

We are not claiming the agent has that idea. The program is built to pay whoever does, at once and in full.