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:

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:

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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