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$2^2$ is between the twin primes $3$ and $5$. Are there any other instances of a power of two between twin primes? If so, how many?

That there are Mersenne primes (primes of the form $2^n-1$) makes this a little more tantalizing, but a brief search didn't spit out any results right away.

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No: if $n$ is odd, $3\mid 2^n+1$, and if $n$ is even, $3\mid 2^n-1$.

To see this, note that $2^n+1=2^n+1^n$, which is divisible by $2+1$ if $n$ is odd, while $2^{2k}-1=4^k-1^k$ is divisible by $4-1$.

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  • $\begingroup$ @Jedediyah: You’re welcome! $\endgroup$ – Brian M. Scott Feb 21 '16 at 21:50
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    $\begingroup$ Alternatively, $2^n-1,2^n,2^n+1$ are three consecutive integers. One of them has to be divisible by $3$, and it can't be $2^n$. $\endgroup$ – f'' Feb 21 '16 at 23:57
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    $\begingroup$ @f'': True enough; sometimes the first workable idea that occurs to one isn’t the simplest idea! $\endgroup$ – Brian M. Scott Feb 21 '16 at 23:59
  • $\begingroup$ Ah yes @f''! I recognize that now. I think I have seen this problem before but worded differently and did not make the connection until now :) $\endgroup$ – Carser Feb 22 '16 at 3:23
  • $\begingroup$ I don't understand how it helps to look at $2^n + 1$ as $2^n + 1^n$, i.e. how it's more clear that way that it's divisible by $2 + 1$ if $n$ is odd. Same for the second case. $\endgroup$ – Cedric Reichenbach Feb 22 '16 at 8:56
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More complicatedly:

  • If $2^n-1$ is a (Mersenne) prime, then $n$ must be prime: if $p|n$, then $2^p-1|2^n-1$.
  • If $2^n+1$ is a (Fermat) prime, then $n$ must be a power of $2$: if $n=pk$ with $p$ odd, then $2^k+1|2^n+1$.

The only $n$ which is both prime and a power of $2$ is $2$ itself. So the only twin primes surrounding a power of $2$ will surround $2^2=4$.

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    $\begingroup$ Ohhhh I like it. $\endgroup$ – Carser Feb 22 '16 at 3:38
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Just some basics. Twin primes use the last digits of 1 & 3, 7 & 9, and 9 & 1 5 is and exception to the general rule. The powers of 2 use the last digits of 2, 4, 6, and 8. The only valid powers of 2 will have the last digits of 2 and 8. 2^2 = 4 and 2^3 = 8. Lots of luck trying to find more of them.

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