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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.

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    $\begingroup$ Well, try an example or two. It only takes one counterexample to disprove a claim. $\endgroup$
    – lulu
    Aug 10, 2022 at 16:13
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    $\begingroup$ Write $2^n+2^m=2^n(1+2^{m-n})$ and note that the second factor is not a power of $2$. So it is never power of two for $m>n$. $\endgroup$ Aug 10, 2022 at 16:14
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    $\begingroup$ the argument provided by @DietrichBurde Shows that it is never the case that you get a power of $2$ this way (trusting that $n\neq m$). $\endgroup$
    – lulu
    Aug 10, 2022 at 16:18
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    $\begingroup$ The proof is trivial. We may assume that $m>n$ by symmetry. Then $1+2^{m-n}$ is odd, and we are done. $\endgroup$ Aug 10, 2022 at 16:19
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    $\begingroup$ The argument is not that $2^n+2^m$ is even. It is that $2^n(1+2^{m-n})$ cannot be a power of two, because $1+2^{m-n}$ is odd. $\endgroup$ Aug 10, 2022 at 16:22

1 Answer 1

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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?

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  • $\begingroup$ Yes , it helped me. $\endgroup$
    – Get_ Maths
    Aug 10, 2022 at 17:11
  • $\begingroup$ Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. $\endgroup$ Aug 10, 2022 at 17:22
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    $\begingroup$ @BillDubuque If I had known this was a dupe answer to a dupe FAQ, I would not have posted it. $\endgroup$ Aug 10, 2022 at 17:23
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    $\begingroup$ @BillDubuque, what do you recommend for someone whose chief reason for being here is to help people who are asking for help, in real time when possible? Should I just channel that energy into linking them, in a friendly, welcoming, non-scolding way, to a high-quality previous instance of the question? The tension regarding site goals is getting to where I want less and less to do with this site. Can't we just do away with rep, beyond a certain level, or something? $\endgroup$ Aug 10, 2022 at 17:52
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    $\begingroup$ I can certainly empathize, since I felt similarly in the past. Although it might not be obvious at first glance, it can be very rewarding organizing the site, e.g. it sparks one to think more deeply about questions in order to understand how variations on common exercises can be handled through (slight) generalizations of prior answers. This leads to iterative improvements of prior answers (hopefully eventually to "proofs from the book") and - by increasing knowledge of answerers - to better answers to future questions. None of this occurs if we continually repost dupe answers. $\endgroup$ Aug 10, 2022 at 18:22

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