## Have a power-ful day

by

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 24=16 and 42=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.

Of course $a^b = b^a$ has infinitely many trivial solutions (solutions where $a=b$), 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 $\ln(x)/x$. 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 $(a, b)$ to the exponential equation above, and clearly 2 is the only integer smaller than e that could lead to a nontrivial solution.