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Are there any twin primes of the form $2^n − 1$, $2^n + 1$, for $n > 2$? If so, give an example, and if not, prove there aren’t any.

Hint: $k$, $k + 1$, $k + 2$ is a complete residue system modulo $3$, for any choice of $k$.

I've tried to find an example of twin primes of the form specified above, but I can't seem to find any (simply through guess and check). How can I prove that there are no primes of the above form?

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  • $\begingroup$ Let $n=3$ or $n=6$ (or many others). $\endgroup$ – André Nicolas Jul 3 '16 at 2:18
  • $\begingroup$ I'm sorry! I have edited my question above to state $2^n−1, 2^n+1$ instead of $2n−1, 2n+1$ $\endgroup$ – Crazed Jul 3 '16 at 2:22
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First, note that $2^n$ is not a multiple of $3$ for any $n$. Since exactly one of any three consecutive integers is a multiple of $3$, one of $2^n - 1$ or $2^n + 1$ is a multiple of $3$.

But $2^n \pm 1 > 3$, since $n > 2$. Thus the multiple of three is not $3$, and hence not prime.

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