Today is 2/4/16 or 4/2/16, depending on where you live and how you write dates. Either way, it’s a great day because 2^{4}=16 and 4^{2}=16. There aren’t many days like that (although we are treated to two this year), so it’s worth taking a moment to celebrate.

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February 4, 2016 at 11:11 am |

Of course has infinitely many trivial solutions (solutions where ), but it also has infinitely many nontrivial solutions. However, the only nontrivial integer solution is the pair of numbers 2 and 4.

The usual approach to this problem is to examine the function . Its maximum occurs at e, and for all other positive values of x, it is a two-to-one function. That two-to-one behavior corresponds to solutions to the exponential equation above, and clearly 2 is the only integer smaller than e that could lead to a nontrivial solution.

May 23, 2016 at 7:43 am |

Interesting analysis challenge: Show that no continuous function is two-to-one.