For example 11 and 101 are primes, but apart from them, can we determine how many primes on the form $100\cdots 00 1$ there exist (in decimal number system)?
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1I think that it is an open problem. Note that $10^n+1$ is prime implies that $n$ is a power of $2$, though. – ajotatxe Oct 09 '18 at 18:34
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1Related. – JMoravitz Oct 09 '18 at 18:36
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the ones permitted to be prime are $10^{2^n} + 1.$ No guarantees, $10001 = 73 \cdot 137$ – Will Jagy Oct 09 '18 at 18:36
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Easy to formulate your problem.....Absolutely another thing is try to solve it. – Ataulfo Oct 09 '18 at 18:39