Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$?

prime-numbers

Solution 1:

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 $3\le m \le 8195$.

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