Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$?
I confirm there isn't such prime for $n < 64$.
Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$?
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prime-numbers
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0We know if $n$ has an odd prime factor, the result is composite. So the only possibilities are for $n$ a power of two. These grow rapidly enough that heuristic arguments suggest an "expected" number of these primes will be small. – 2017-01-22
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1These are also a special case of _generalized Fermat primes_ and you can find a little more information about them under that name. – 2017-01-22
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0That $10^{2^{10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}}}+1$ is not prime is almost certain but it's likely to be unprovable. – 2017-01-22
1 Answers
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Note that $(10^{n}+1)|10^{n(2k+1)}+1$ for $n$, $k\in \mathbb{N}$.
According to the Table $\boldsymbol{1}$ (page $24$ or $\frac{30}{55}$ of the pdf) from this journal:
$10^{2^{n}}+1$ has no (known?) prime factors for $n=13$, $14$, $21$, $23$, $24$, $25$, $\ldots $
That means $10^m+1$ is composite continually for $3\le m \le 8195$.