33 as the sum of three cubesSource: Pixabay

The number line as we all learned in school, looks like it should be a pretty sedate place, but it is actually roiling with interesting problems.

One of those problems is called the sum of three cubes problem. It asks whether any whole number on the number line, both positive and negative, can be expressed as the sum of three cubes in the form:

k = x³ + y³ + z³, where k is a whole number.

The numbers 29 and 26 can be expressed like this:

29 = 3^{3} + 1^{3} + 1^{3}

26 = 114,844,365^{3} + 110,902,301^{3} + –142,254,840^{3}

### Stubborn Numbers

Some numbers on the number line will never have a solution, such as all whole numbers that have a remainder of 4 or 5 when they are divided by the number 9. While the answers for many numbers have been known for years, the number 33 has remained a mystery for 64 years!

However, just recently, a mathematician at the University of Bristol named Andrew Booker took a crack at a solution, and he discovered that:

33 = 8,866,128,975,287,528^{3} + –8,778,405,442,862,239^{3} + –2,736,111,468,807,040)^{3}

Booker solved the problem by creating an entirely new search algorithm, which he ran on a university's supercomputer for three weeks straight. Originally, Booker had thought the solution would take six months to solve.

Booker's algorithm is a more efficient way of locating solutions, and according to Booker, it runs "maybe 20 times faster" than earlier algorithms have run. Still, Booker had to search numbers all the way up to plus and minus 10^{16}, or ten quadrillions, before finding the answer.

Until Booker found the solution, 33 was only one of two integers below the number 100 that couldn’t be expressed as the sum of three cubes. Now, the only remaining number is 42. Booker has already determined that no solutions for 42 exist in the 10^{16} range, so he'll have to look in the 10^{17} range.

Number theorists already know that there are 11 integers between the numbers 101 and 1,000 that don't have a sum-of-three-cubes solution: 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975.

### "The answer to the ultimate question of life, the universe and everything is 42."-- Doublas Adams, The Hitchhiker's Guide to the Galaxy

As for the number 42, it has become famous since Douglas Adams wrote in *The Hitchhiker's Guide to the Galaxy*: "The answer to the ultimate question of life, the universe and everything is 42." Both Google's headquarters and CERN have office complexes named for the number. The address 42 Wallaby Way appears on the diving mask in Pixar's Finding Nemo.

Adams before his death in 2001, only revealed the secret of the number 42 to his friend, actor Stephen Fry, who claims he'll take the secret to his grave. Maybe Booker should proceed a little carefully around the number 42.