Subject: Re: A diophantine equation
Date: Fri, 12 Sep 1997 15:34:38 -0400
From: Alex Bogomolny
The gist of the problem is that exponential functions grow faster than power functions: for n>n0 kn>nm, where n0 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, abcde>a2e. By inspection, a2e>e5 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?