No power of 2 can be written as the sum of two or more consecutive positive integers.

The proof of this statement is by contradiction. Let us assume that it is possible to write a power of 2 as the sum of two or more consecutive positive integers. Then, we would have two positive integers \(m\) and \(n\) such that: \[m+(m+1)+\ldots+(m+n)=(2m+n)(n+1)/2=2^k.\]

We can simplify this into \((2m+n)(n+1)=2^{k+1}\)

The right side of this equation has no odd factors, but at least one of \(2m+n\) and \(m+1\), which are both greater than 1, is odd. Therefore, the above equation cannot hold, and the initial assumption was incorrect.

In other words, if we could write a power of 2 as the sum of two or more consecutive positive integers, then we would have an odd number equal to an even number. This is a contradiction, so the initial assumption must be false.