Each of A and N is a positive integer, and P is a prime number that satisfy this equation:

2^P + 3^P = A^N

Can N exceed 1?

If so, give an example. If not, prove that no positive integer greater than 1 is possible.

Ignore P's primality for a moment.       
1. if P is even, then LHS = {3,7} mod 10; these values of n need not further detain us.       
2. If P is odd, then LHS = 5, mod 10, so we seek divisibility by 5.       
3. Checking small values, it is at once obvious that LHS is singly divisible by 5, unless P(prime or not) = (10n-5). Even then, the power of 5 can at most increase only by increments of 1       
for values n = 5^(m+1)/2 {3,13,63, etc} i.e. where 10n-5 is itself a power of 5. This rate is far less than the increases in P.       
4. Be that as it may, the only prime of form (10n-5) is 5. 2^5+3^5=275 is not a power for n>1, so we are done.

Edited on December 16, 2010, 2:01 am

Posted by broll on 2010-12-14 15:08:55