# 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: $x^3 + y^3 + z^3 = 33$ and $x^3 + y^3 + z^3 = 42$. 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:

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

\$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 $x^3 + y^3 + z^3 = n$ for all positive integers $n$ less than 100. For now, the smallest value of $n$ for which the existence of a solution is not known is $n = 114$.

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

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