Show that if aⁿ+bⁿ= 2^m, and a, b, m, and n are positive integers (n>1), then a=b.

Divide the equation by the power of 2 that a and b share.

If the right-hand side of the reduced equation = 2, then (a,b,n,m) = (2,2,n,n+1) is a solution of the original equation.

Assume the right-hand side of the reduced equation >=4.  Then the 2 terms of the left-hand side can't both be even, since we've factored out common powers of 2, and one can't be odd and the other even, since that would yield a odd sum, so they're both odd.  But then their sum = 2mod4, which is impossible. 

 

Posted by xdog on 2007-04-09 08:27:10