After cracking the ‘sum of cubes’ puzzle for 42, researchers discover a new solution for 3

After cracking the ‘sum of cubes’ puzzle for 42, researchers discover a new solution for 3 by Jennifer Chu , Massachusetts Institute of Technology

In September 2019, researchers, harnessing the combined power of half a million home computers around the world, for the first time found a solution to 42. The widely reported breakthrough spurred the team to tackle an even harder, and in some ways more universal problem: finding the next solution for 3. Credits: Christine Daniloff, MIT

What do you do after solving the answer to life, the universe, and everything? If you’re mathematicians Drew Sutherland and Andy Booker, you go for the harder problem.

In 2019, Booker, at the University of Bristol, and Sutherland, principal research scientist at MIT, were the first to find the answer to 42. The number has pop culture significance as the fictional answer to “the ultimate question of life, the universe, and everything,” as Douglas Adams famously penned in his novel “The Hitchhiker’s Guide to the Galaxy.” The question that begets 42, at least in the novel, is frustratingly, hilariously unknown.

In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, 42, had similarly eluded mathematicians for decades. The equation x3+y3+z3=k is known as the sum of cubes problem. While seemingly straightforward, the equation becomes exponentially difficult to solve when framed as a “Diophantine equation”—a problem that stipulates that, for any value of k, the values for x, y, and z must each be whole numbers.

When the sum of cubes equation is framed in this way, for certain values of k, the integer solutions for x, y, and z can grow to enormous numbers. The number space that mathematicians must search across for these numbers is larger still, requiring intricate and massive computations.

Over the years, mathematicians had managed through various means to solve the equation, either finding a solution or determining that a solution must not exist, for every value of k between 1 and 100—except for 42.

In September 2019, Booker and Sutherland, harnessing the combined power of half a million home computers around the world, for the first time found a solution to 42. The widely reported breakthrough spurred the team to tackle an even harder, and in some ways more universal problem: finding the next solution for 3.

Booker and Sutherland have now published the solutions for 42 and 3, along with several other numbers greater than 100, this week in the Proceedings of the National Academy of Sciences