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Does there exist a prime number $p$ such that there exist no prime numbers between $p$ and $p^n$, $n\geq2$, $n\in N$? If yes, what is the smallest such prime number?

Bill Dubuque
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Pragnya Jha
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  • We are not here to blindly answer your homework questions. Please add details about what you think or have already tried, or add some details of the techniques you think might be useful, so we can write answers at the right level of knowledge for you – lioness99a Feb 10 '20 at 09:08
  • This is not a homework question. This is a question I came up with myself while I was studying through the first chapter of Hardy's elementary number theory book. I couldn't make any headway and hence, asked this question. I will delete the question, if it is inappropriate. – Pragnya Jha Feb 10 '20 at 09:10
  • The way you have asked it, it reads like a homework question. People will only put effort in to answer if you can show that you have thought about the question yourself first. You can edit your question and add some more details – lioness99a Feb 10 '20 at 09:11
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    Look up Bertran's postulate – pancini Feb 10 '20 at 09:16
  • Thank you, mate! The postulate quite easily solves my problem. – Pragnya Jha Feb 10 '20 at 09:38

1 Answers1

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Bertrands postulate says there's always a prime between $n>2$ and $2n-2$ this can actually give statements like:

There exists a prime between $x^2$ and $(x+1)^2$ or (inclusive, letting $x=2j+1$ ) there exists a prime between $2j^2+4j+3$ and $x^2$ but this is mostly playing around trying to simulate Legendre's conjecture with Bertrands postulate.

In any event, No there doesn't exist a prime like you wanted in fact Bertrand implies there are $\lfloor n\log_2(p)\rfloor-\lfloor \log_2(p)\rfloor$ primes in that interval at the least.