Finding meeting point of discreet functions [added context per advised] Hi, I am self-studying number theory, using text of aata, Beezer, 14 and elementary number theory, Pan, 91. So far I have finished group theory but not yet ring and field. I encountered exercise as below in 2 (pg 261.) Appreciate explanation that utilize group theory. Thanks. [rewording per required] I have a sum of arithmetic series $n(1+n)/2$, and a linear function $nk$. If I want to find the meeting point of their sum with $2^m$. Is there a way better than brutal force try n error? Thank you.

You can factor out n:

$2^m=n\cdot \left( \frac{1+n}{2}+k \right)$

n has to be a faktor of $2^m$. Let it be $2^p$.

$2^m=2^p\cdot \left( \frac{1+2^p}{2}+k \right)$

multiplying the equation by 2.

$2^{m+1}=2^p\cdot \left(1+2^p+2k \right)$

The equation does not hold, because $1+2^p+2k$ is an odd number.

_Referring to the comment of robjohn:_

If $2^p=1$ and $k=2^m-1$ the equation becomes $2^{m+1}=1\cdot \left(1+1+2^{m+1}-2 \right)$ The equation holds in this case.