Date: Fri, 12 Sep 1997 15:34:38 -0400

From: Alex Bogomolny

Dear Ronald:

The gist of the problem is that exponential functions grow faster than power functions: for n>n_{0} k^{n}>n^{m}, where n_{0} depends on k and m.

a can't be 1 for then all b,c,d, and e must equal 1 but they should be different.
Therefore a is at least 2. Since b,c,d,e must be different, I'll assume for a while that each
is greater than 1. Let m = 5. By assumption, a^{bcde}>a^{2e}. By
inspection, a^{2e}>e^{5} for e>7. Therefore, the original problem
can't have solutions with e>7. Similarly you can establish that there is no solutions
with d>5, c>2, b>2. Since b and c are different one of them is, at best, 1 and another
is 2. Thefore, both d and e must be no less than 3. In all, you have fewer than 2*(7-2)*(5-2) = 30
quadruples (b,c,d,e) to test. Can you take it from here?

Regards

Alexander Bogomolny