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The List of unsolved problems in mathematics contains varies conjectures of exotic primes like:

to name just a few. Each of these primes is provided with special property. Sitting enthroned over the list on the wiki page you'll find the Twin Prime Conjecture (ignoring Catalan's Mersenne conjecture for the moment). So here is my

Question: Assuming the infinitude of all exotic primes, for which is it possible to disprove the infinitude of twin prime pairs containing one exotic prime?


Example

It could be done for e.g. primes of the form $p_n=(6n)^2+1$, where it's obvious to show that $p_n\pm 2$ are composite and $p_n$ is called isolated prime. And the weaker $5$th Hardy-Littlewood conjecture asserts that $a^2+$1 is prime for an infinite number of integers $a>1$ [from Bouniakowsky Conjecture].

Don't get me wrong: This is not what I'm looking for! The question is on the primes given in the list above: Fermat/SophieGermain/Mersenne/... and how to disprove that they are infintely often one of the twins in a pair. Mea culpa, if this is misleading.

Collected partial results

  • Percentages given in the list above, show the ratio of exotic primes having a twin (some where counted twice, specially in the regular case).

  • For Fermat primes it seems promising ($100\% !$) to prove that every prime has a twin, but then this restricts to Fermat primes of the form $2^{2^{2n}}+1$, since $7 \mid 2^{2^{2n+1}}+3$ (see coment below) and $2^{2^{2n+1}}-1$ is obviously composite.

  • An analogous analysis might done for Mersenne primes, but I haven't yet.

  • The Wiki page on Twin Primes gives some more general ways to tackle the problem, but I'm not sure, if they are really useful:

    1. Every twin prime pair except $(3, 5)$ is of the form $(6n - 1, 6n + 1)$ for some $n$, and with $n \neq 1$, $n$ must end in $0, 2, 3, 5, 7 \text{ or } 8$. -- This seems related to my example given above, since $1,4,6$ and $9$ are missing, which show up as end digits of square numbers, see here. $0$ and $5$ seem to be exceptions.

    2. The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$.

So if you feel that you can disprove the twin prime conjecture on any of these exotic primes, I'd be every so happy to read your answer here. If you think you can prove it for a kind of primes, where the infinitude is also proven, send me an eMail.

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    $7 \mid 2^{2^{2n+1}}+3$ can be written as $2^{2^{2n+1}}-4 \equiv 4\left(2^{2(4^n-1)}-1\right)\equiv 0\bmod 7$. With $4^n-1\equiv 0\bmod 3$ we have $4\left(2^{2\cdot 3k}-1\right)\equiv 0\bmod 7$, proven [here](http://math.stackexchange.com/q/148418/19341).2012-05-23
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    For the interested reader:[Regularities of Twin, Triplet and Multiplet Prime Numbers](http://arxiv.org/abs/1103.0447) by H. J. Weber2012-05-24
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    What are the percentages in your list of exotic primes?2018-05-25
  • 0
    @Vincent: *Percentages given in the list above, show the ratio of exotic primes having a twin...*2018-05-29
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    Interesting. The percentages are uncannily large. Perhaps because they are only calculated over small primes?2018-05-29

2 Answers 2