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[Migrated from Math Stack Exchange]

More than a year ago, I posted the following on the Math Stack Exchange.

Consider $2^n-1$. Based on checking a few small numbers for $n$ (in fact, the first ten natural numbers) we might "conclude" that "$n$ is prime if and only if $2^n-1$ is prime." However, further investigation shows that for $n=11$, $2^n-1$ is not prime. Thus our claim that "if $n$ is prime then $2^n-1$ is prime" is a near miss. However, we can prove its reverse, that "if $2^n-1$ is prime then $n$ is prime". I am looking for some elementary number theoretic examples that both directions of our eventually fake biconditional are near misses.

To my great surprise, I haven't received even a near miss comment, let alone an answer. I know that the notion of "near miss" is disputable. Thus you might want to ignore my example above, and give an example of a biconditional statement (number-theoretic or not, historical or not, made-up or not) that seems to be true for some reasons (e.g., observing a number of cases, or intuition, etc) in both directions, but then it turns out that it is, in fact, false in both directions.

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    $\begingroup$ Your original example is quite contentious. To date there are only around 40 known Mersenne primes, and a much larger number of prime numbers. Thus, it is likely true that for asymptotically 100% of all primes $p$, the number $2^p - 1$ is not prime... so I don't understand what you mean by 'near miss'. $\endgroup$
    – Stanley Yao Xiao
    Commented Nov 29, 2017 at 16:03
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    $\begingroup$ @StanleyYaoXiao That is exactly why I wrote "you might want to ignore my example" :) Please ignore it. I needed to mention it since it was in the original post. Please read just the last three lines of the post :) $\endgroup$
    – Amir Asghari
    Commented Nov 29, 2017 at 16:06
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    $\begingroup$ @AmirAsghari: it seems like this question should be community wiki because there could be multiple good answers --- as I understand it, as the OP you can make that change. $\endgroup$
    – Nik Weaver
    Commented Nov 29, 2017 at 18:06
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    $\begingroup$ @NikWeaver Actually, a question can be changed to CW by moderators, not by the OP. (A poster can make an answer CW, but the same is not true for questions. Here is somewhat related discussion on meta: Community Wiki in the hands of moderators.) $\endgroup$
    – Martin Sleziak
    Commented Nov 29, 2017 at 18:22
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    $\begingroup$ Your reputation on MO is high iff you're a strong mathematician. $\endgroup$
    – Christian Remling
    Commented Nov 29, 2017 at 18:46

1 Answer 1

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False claim: A Hausdorff topological space is compact if and only if it is sequentially compact.

It's believable if your intuition of Hausdorff spaces comes entirely from metric spaces (where the claim is, in fact, true). However, both directions are false for different reasons:

Counterexample I: the Alexandroff line is sequentially compact but not compact.

Counterexample II: the (Tychonoff) product of $|\mathbb R| $-many copies of the unit interval is compact but not sequentially compact.

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    $\begingroup$ Thank you, Adam, you made my day and the question crystal clear :) $\endgroup$
    – Amir Asghari
    Commented Nov 29, 2017 at 16:28
  • $\begingroup$ Counterexample I can be simplified, it's enough to take $\omega_1$ with order topology. Similarly, I believe in II it's enough to take a product of two-point sets, but I suppose one can argue as to whether this one is really simpler. $\endgroup$
    – Wojowu
    Commented Nov 30, 2017 at 9:44

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