Prove that the only positive integers z which can be expressed as a sum of at least two consecutive positive integers are those such that z =/=2ⁿ , for any natural number n.

The sum of a series of consecutive integers is the product of the number of terms and the "middle value" (the middle term, or the average of the two middle terms). In order for this product to be a power of 2, then both the middle value and the number of terms must be a power of 2. But if there are at least two terms and the number of terms is a power of 2 then there are an even number of terms, so the middle value is not even an integer much less a power of 2.

Oh, I forgot the other half of the proof. If z is not a power of 2 then it has an odd prime factor, n. Let n be the number of terms, and let z/n be the middle value. So any number, z, that is not a power of 2 can be expressed as a sum of consecutive integers.