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: