Problem:
Find the number of positive integers less than that can be expressed as the difference of two integral powers of .
Solution:
Assume that positive integer can be represented as for integers and . Because must be a positive integer which implies that is nonnegative.
If for nonnegative integers , and with and , then the greatest power of 2 that divides the left side is , while the greatest power of 2 that divides the right side is . Hence and . Therefore no positive integer can be represented as the difference of two integral powers of 2 in two distinct ways.
If , then , so .
Therefore there are positive integers that can be expressed as a difference of two integral powers of 2 .