Prove : $2^n + 2^m ≠ 2^p$
For any $n,m$ ($n≠m$) and $p$ being positive integers.
I hadn't yet studied very much mathematics yet but I came across it when I had to build a python code. Any help is appreciated.
Prove : $2^n + 2^m ≠ 2^p$
For any $n,m$ ($n≠m$) and $p$ being positive integers.
I hadn't yet studied very much mathematics yet but I came across it when I had to build a python code. Any help is appreciated.
You might enjoy this view of the question, which provides an alternative proof.
Writing $2^m$ in binary, we have a $1$ followed by $m$ $0$'s. For example:
$2^1: 10 \\ 2^5: 100000$
How can we add two numbers of this form, and obtain a third one of the same form? If the $1$'s in each number are in different locations, then we get a number with two $1$'s in it:
$10 + 100000 = 100010$
The only way to get a sum with only a single $1$ in it is if the $1$'s in the two summands line up, giving a $0$ and a "carry" digit:
$10 + 10 = 100$
Does this help?