Annual Report 2019

The Sum of Three Cubes

Simons Collaboration on Arithmetic Geometry, Number Theory and Computation

In the fall of 2019, Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology and one of six principal investigators with the Simons Collaboration on Arithmetic Geometry, Number Theory and Computation, together with his collaborator Andrew Booker, a mathematician at the University of Bristol, published an equation that appealed to both number theorists and fans of The Hitchhiker’s Guide to the Galaxy.

–80,538,738,812,075,9743 + 80,435,758,145,817,5153 + 12,602,123,297,335,6313 = 42

The question of whether a whole number can be written as the sum of three cubes might not generally be “the ultimate question of life, the universe, and everything,” but it is surprisingly tough to answer. And while it may seem like a curiosity, it has deep connections to important areas of research in number theory and algebraic geometry.

Booker became interested in the sum-of-three-cubes problem after watching a video about it on the popular YouTube math channel Numberphile. In 2015, he was perusing the channel looking for activities to do with the math club at his children’s school. He ran into a video, coincidentally featuring his colleague Tim Browning, about the sum-of-three-cubes problem. Browning noted that at the time 33 was the smallest integer for which the problem was unresolved.

It is fairly easy to convince yourself — or, in Booker’s case, an elementary school math student — that a number that has a remainder of 4 or 5 when divided by 9 cannot be written as the sum of three cubes. (Every integer can be written in the form 3n, 3n + 1, or 3n − 1 for some n. If you cube those expressions, you see that every cube is within 1 of a multiple of 9. Add them up, and it’s clear that any sum of three cubes must be within 3 of a multiple of 9.)

In February 2019, Booker used some clever coding and about a week’s worth of time on his university’s computing cluster to find three cubes that sum to 33. The next month, at a conference, he asked Sutherland if he wanted to join the search for three cubes that sum to 42, which was now the smallest number for which the answer was unknown. Sutherland had experience with large parallel computing projects and expertise in the algorithms and programming languages they might use to tackle the problem. Later that year, they used Charity Engine, a crowdsourced network of computers, to show that 42 can be written as the sum of three cubes. That computation finished off the two-digit numbers, leaving 114 as the smallest number for which the problem is open.

Their work extends to a related question: If a whole number can be written as a sum of three cubes, how many ways can it be so written? The number 3 can be written as 13 + 13 + 13, but it can also be written as 43 + 43 + (−5)3. Both of those representations are fairly easy to find just by playing around with arithmetic. But in 1992, mathematician Roger Heath-Brown conjectured that any number that can be written as a sum of three cubes can be written as a sum of three cubes in infinitely many ways. Sutherland and Booker discovered another representation of 3 as a sum of three cubes, providing one more piece of evidence that Heath-Brown’s conjecture is correct.

The sum-of-three-cubes question lies on what Sutherland describes as the “interesting frontier between what we can do theoretically and what can be turned into a practical computation.” This frontier is the sweet spot for the collaboration, one of whose primary objectives is to build computational tools that can help feed into theoretical understanding of number theory and arithmetic geometry.

A graph showing the computational muscle needed to discover a new representation for 3 as a sum of three cubes. The computation required nearly 4 million hours of CPU time across more than 400,000 PCs. Each dot in the graph represents 50 PCs. The y-axis represents the number of computing hours, and the x-axis represents a key parameter d that partitions the search space. Values of d with no large prime factors are colored blue, while those with a large prime factor are colored purple. Credit: Andrew Sutherland

In this current work, Sutherland was able to adapt algorithms he had previously used to compute zeta functions and L-functions — two important tools used to study problems like the Riemann hypothesis and Sato–Tate conjecture — to the sum-of-three- cubes problem. He is optimistic that the work he and Booker did to speed up the code for this project will have rewards in the world of zeta and L-functions. “There’s a nice synergy there,” Sutherland says.

Although the sum-of-three-cubes problem is simple enough for even young students to understand, it is related to more abstract theoretical work done by members of the collaboration and other mathematicians in the field. For decades, number theorists have pushed the discipline forward by translating problems about discrete entities (whole numbers) into questions about continuous objects (algebraic varieties, which are solution sets to polynomial equations). The sum-of-three-cubes question is another one of these problems. Previous work on the question done by Harvard University mathematician Noam Elkies, also a principal investigator in the Simons collaboration, translates the problem into the question of finding rational points on cubic surfaces, which are defined by polynomials of degree 3.

The Numberphile YouTube channel explores the discovery of a new way to represent 3 as a sum of three cubes.

Every time Booker and Sutherland manage to write a new number as the sum of three cubes, they fit one small piece into the puzzle of which numbers may be written that way. All the same, the puzzle may never be completely solved. The two collaborators’ work is just one part of a more general question researchers have faced for decades: Which polynomial equations have integer solutions? In 1970, mathematicians showed that there is no algorithm that can answer that question for all polynomials.

Collaboration researchers may find a way to answer the question fully for the sum-of-three-cubes problem using theoretical rather than computational tools — 
or they may not be able to. Far from making Booker and Sutherland feel hopeless, this prospect motivates them to redouble their efforts at approaching the problem from an algorithmic point of view. “It may turn out that our only tool for getting at these kinds of questions is by running computations,” Sutherland says.