Sum of Three Cubes

I now have a new example of an existence proof to show my students.

Last year, mathematicians Andrew Booker and Andrew Sutherland found solutions to the following two equations: and . The first was found by Booker alone; the latter was found by the collaboration of both mathematicians. These deceptively simple-looking equations were cracked with a lot of math and a lot of computational firepower. The solutions:

$latex (–80,538,738,812,075,974)³ + 80,435,758,145,817,515³ + 12,602,123,297,335,631³ = 42$

At the time of this writing, that settles the existence of solutions of for all positive integers less than 100. For now, the smallest value of for which the existence of a solution is not known is .

For further reference, including links to the original articles by Booker and then Booker and Sutherland, please see:

• https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326
• https://www.livescience.com/diophantine-42-solved-meaning-of-life.html