Is it ever the case that 2^m+3^m is a perfect power, whenever m is a positive prime number?

No.

m=2 does not yield a prime.

For m=4k+1, the terminal digit of 2^m is 2 and of 3^m is 3.

For m=4k+3, the terminal digit of 2^m is 8 and of 3^m is 7.

In both cases the terminal digit of the sum is 5.  Since sum > 5, sum is never prime.

 

 

 

Posted by xdog on 2007-06-25 08:45:46