Theorem
An integer can be expressed as a sum of 2 or more positive integers if and only if is not a power of 2.
Proof
Part 1: Forward Direction
Assume is not a power of 2. We find consecutive integers starting at with length such that their sum is .
Note:
Thus, we find such that or, identically,
Note that and have opposite parity (difference is odd), so we must decompose (and thus ) into an even and odd part. To do this, we take out successive factors of 2 such that an odd part remains. Hence, with nonnegative integer and odd integer , which is guaranteed to exist as is not a power of 2. Let where .
We now decompose by assigning and to and without introducing negative integers.
Case 1:
Let , . and is a nonnegative integer as , so .
Hence , as desired.
Case 2:
Let , . and is a nonnegative integer as .
Hence, , as desired.
Part 2: Reverse Direction
Assume to the contrary is a power of 2, for some positive integer .
Since and the two terms and have different parity, there must be an odd factor of and thus for it to be expressed as a sum of 2 or more positive integers. As , there are no odd factors so this expansion is impossible.