**Question:**

Prove that it is impossible to represent the number $ 2 ^ n, n \in N $, the sum of two or more consecutive natural numbers.

**Answer**

Suppose we can express the number $ {2 ^ n}, n \in N $ into the sum of m consecutive natural numbers: $ {2 ^ n} = k + (k + 1) + … + (k + m – 1) $ (1)

For $ k, m \in N, k \ge 1, m \ge 2 $

We have:

$ \begin { array} {l}

(1) \Leftrightarrow {2 ^ n} = \frac {{m (2k + m – 1)}} {2} \Leftrightarrow

& {2 ^ {n + 1}} = m (2k + m – 1)

\end {array} $

m and $ \left ({{\rm {2k}} + {\rm {m} } – {\rm {1}}} \right) $ is greater than 1.

On the other hand:

$ \left ({{\rm {2k}} + {\rm { m}} – {\rm {1}}} \right) – {\rm {m}} = {\rm {2k}} – {\rm {1}} $, odd so 2 number m and $ \left ({{\rm {2k}} + {\rm {m}} – {\rm {1}}} \right) $ has an odd number> 1 $ \Rightarrow {2 ^ {n + 1}} $ divisible by an odd number> 1. Ridiculous

So we cannot represent $ {2 ^ n}, n \in N $ as the sum of two or more consecutive natural numbers