Show that if N = 2^k, it is impossible to express N as a sum of two or more consecutive positive integers.?

Question

Derek C, I appreciate the help.
Does your answer consider the possibility of three consecutive positive integers? Four?
Thanks so much

Answer 1

Let's consider 2 cases:
First, look at 1+2+... + n= (n)(n+1)/2 = 2^k
This gives n(n+1) = 2^(k+1) (*)
Since there are at least 2 consecutive integers here,
n >1, so one of n and n+1 is odd and so must
contain an odd prime factor(possibly itself)
so (*) is impossible.
Now look at
x+1 + x+2 + ... + x+n = 2^k
This gives
nx + n(n+1)/2 = 2^k
or
2nx + n(n+1) = 2^(k+1).
or
n(2x + n+1) = 2^(k+1) (**)
Again, n> 1 and n is a divisor of a power of 2
So n is a power of 2.
But then n+1 is odd, 2x + n + 1 is odd
and again it must contain an odd prime factor.
So (**) is impossible.
Hope that helps!

Answer 2

Hello

in response to your question, yes my proof does consider any number of consecutive integers as I've used an arbitrary number x to represent that in my proof. I've also chosen an arbitrary first term n.