  Using completely elementary methods, we find all powers of 3 that can be
written as the sum of at most twenty-two distinct powers of 2, as well as all
powers of 2 that can be written as the sum of at most twenty-five distinct
powers of 3. The latter result is connected to a conjecture of Erd\H{o}s,
namely, that 1, 4, and 256 are the only powers of 2 that can be written as a
sum of distinct powers of 3.
  We present this work partly as a reminder that for certain exponential
Diophantine equations, elementary techniques based on congruences can yield
results that would be difficult or impossible to obtain with more advanced
techniques involving, for example, linear forms in logarithms.
