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I was just playing with $10^n + 1$, and the I realize that I found only 3 of them ($n=0,1,2$)

I was wondering whether there's another $n$ in which $10^n + 1$ is prime, or is this not yet known? My limited programming skill only check up to $n = 30$
Tensor
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    For composite $n \ge 3$, the number $10^n+1$ is not prime. If $a$ divides $n$, then $10^a+1$ divides $10^n+1$. – Crostul Jul 17 '23 at 08:09
  • What about when $n$ is prime? – Tensor Jul 17 '23 at 08:20
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    Another prime of this form must be huge since $n$ must be a power of two and small factors are known upto a very large limit. – Peter Jul 17 '23 at 08:21
  • So, not yet known? – Tensor Jul 17 '23 at 08:22
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    here you can look zp the factorizations of $10^{2^k}+1$. The smallest unknown case is $k=21$ , so another prime of this form must have at least $2$ million digits, so I am pretty sure that no other prime of this form is known. – Peter Jul 17 '23 at 08:27
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    @Crostul Don't you mix this with the generlized Mersenne numbers ? If $n$ is an odd prime , then $10^n+1$ is divisible by $11$ – Peter Jul 17 '23 at 08:37
  • @Peter I think the pieces of information you shared already give everything that the community knows, so might make an adequate answer? I would be delighted to be proven wrong as I know very little, but anyway :-) – Jyrki Lahtonen Jul 17 '23 at 08:57
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    $11$ and $101$ are the only prime terms up to $n=100000$ (see https://oeis.org/A000533) – kabenyuk Jul 17 '23 at 08:59
  • this is an even better site : It states that the smallest unknown case is $k=31$ , so another prime of this form would be far far larger than the largest known prime. It seems that there is almost no hope to find another prime of this form. – Peter Jul 17 '23 at 09:21

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