Sums of positive consecutive Integers

$\begin{aligned}\text{Note that ,}&\\ 131072&=2^{17}\\ \text{We can restate the question as }\\ \sum_{i=1}^n i-\sum_ {j=1}^m j &=2^{17} \hspace{4mm}\color{#3D99F6}(n-m)\geq2\\ \implies \dfrac{n\times(n+1)}{2}-\dfrac{m\times(m+1)}{2}&=2^{17}\\ n\times(n+1)-m\times(m+1)&=2^{18}\\ (n^2-m^2)+(n-m)&=2^{18}\\ (n-m)(n+m+1)&=2^{18}\\ \text{Note that,}&\\ n-m\pmod{2}&\equiv n-m+2m\pmod{2}\\ \implies n-m\pmod{2}&\equiv n+m\pmod{2}\\ \implies\text{ one of } (n-m),(n+m+1) &\text{ is odd}\\ \implies (n-m)(n+m+1)&=2^n\times p,\text{where p is odd} \\ \implies (n-m)(n+m+1)&\neq2^{18}\\ \implies2^{17} \text{cannot be expressed } &\text{ as the sum of consecutive positive integers for any string length}\geq2 \end{aligned}$