Have a power-ful day

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balloonsToday 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.

2 Responses to “Have a power-ful day”

  1. TwoPi Says:

    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.

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