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This question is being used as the main reference when people ask for proofs that $\binom{2^n -1}k$ is always odd, see for example here and here. However, the question and its accepted answer claim a binomial coefficient is odd if and only if it's of this form; this is clearly false, for example $\binom62=15$ and $\binom{2n+1}1=2n+1$ for general $n$.

I didn't go through the accepted answer in the question under discussion, but the previous examples show part of the argument is wrong. Furthermore, IMHO the other two questions have significantly better answers to the parity question, so I'd advocate for one of them to be used as the main reference instead.

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    $\begingroup$ Are you sure you've read the first link correctly? What it says and what you've written in your post here are not the same. Specifically, the linked answer shows that $\binom{n}{k}$ is not divisible by $p$ for all $k$ if and only if $n=p^m-1$ while you write about one specific value of $k$. $\endgroup$
    – KReiser
    Jul 9 at 7:42
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    $\begingroup$ I see your point now, @KReiser I'm being sloppy. However, it does seem to me like this is only clear from the the answer. The question is quite ambiguous in this regard. I'll propose an edit in there to fix this. Should I delete this question? $\endgroup$
    – user347489
    Jul 9 at 7:52
  • $\begingroup$ user347489 no need to delete just ask @KReiser to write this in form of answer if he can then you accept that answer or if he can't write the comment in form of answer then you can take help from mods over here regarding the right action $\endgroup$
    – user876009
    Jul 9 at 15:54

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