problem with primes in intervals

rex88 · P versus NP Offline Joined: 12 Sep 2009 Posts: 22

is the following statement true ?

for every integer $n>0$ there is a prime between $10^n$ and $10^n+10^{n-1}+1$

**Posted:** Mon Oct 19, 2009 5:58 am

mavropnevma

A generalization of Bertrand's postulate states that

For any $c > 0$ there exists positive integer $N_c$ such that for any integer $n\geq N_c$
there exists at least a prime between $n$ and $(1 + c)n$ .

The problem is that that $N_c$ may be quite large ...

For your case, since $10^n + 10^{n - 1} + 1 > (1 + 1/10)10^n$ , we can take $c = 1/10$ .

Also see http://en.wikipedia.org/wiki/Bertrand%27s_postulate

EDIT. In reply to the post following: NO (just rough asymptotic value).

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rex88 · P versus NP Offline Joined: 12 Sep 2009 Posts: 22

thanks

is there any method to effectively find this number $N_c$ for fixed $c$ ?

**Posted:** Mon Oct 19, 2009 12:28 pm