33 and the sum of three cubes

You may have seen that 33 can be expressed as the sum of three cubes.

They are (8866128975287528)^3 + (-8778405442862239)^3 + (-2736111468807040)^3 = 33

The solution was discovered by Andrew Booker from the University of Bristol.

Check out his paper.

If I may quote Andrew’s abstract (ahem!)…

Inspired by the Numberphile video “The uncracked problem with 33” by Tim Browning and Brady Haran, we investigate solutions to x^3+y^3+z^3=k for a few small values of k. We find the first known solution for k=33.

In all the excitement, some people mistakenly attributed the finding to Tim Browning (formerly of Bristol but now in Austria), who was helping publicise Andrew’s result. Tim had introduced many people to the problem via the Numberphile video.

A embarrassed Tim has been quick to remedy this, pointing out Dr Booker’s paper.

33 was previously the lowest number for which the sum of three cubes status was unknown.

There are lower numbers which can never be expressed as the sum of three cubes (4, 5, 13, 14, 22, 23, 31, 32 ) but these were known to be impossible because they can be represented as 9k+4 or 9k+5.

Until now, 33 was the lowest number which was unknown.

There is yet to be a formal proof that all numbers (outside the 9k+4 9k+5 category) can be expressed in this way.

For more on this, see the Numberphile video: The Uncracked Problem with 33 - Numberphile

PS: This is the second breakthrough inspired by the 33 video.