Recall that Bertrand's postulate states that for n ≥ 2 there always exists a prime between n and 2 n. Bertrand's postulate was proved by Chebyshev. Recall also that the harmonic series 1 + 1 2 + 1 3 + 1 4 + ⋯and the sum of the reciprocals of the primes 1 2 + 1 3 + 1 5 + 1 7 + ⋯are divergent, while the sum ∑ n = 0 ∞ 1 n p is convergent for all p &gt; 1. This would lead one to conjecture something like:For all ϵ &gt; 0, there exists an N such that if n &gt; N, then there exists a prime between n and ( 1 + ϵ ) n.Question: Is this conjecture true? If it is true, is there an expression for N as a function of ϵ?

Question

Recall that Bertrand&#039;s postulate states that for   n  ≥  2 there always exists a prime between   n and   2  n. Bertrand&#039;s postulate was proved by Chebyshev. Recall also that the harmonic series  1  +      1    2    +      1    3    +      1    4    +  ⋯and the sum of the reciprocals of the primes      1    2    +      1    3    +      1    5    +      1    7    +  ⋯are divergent, while the sum      ∑          n      =      0              ∞            1          n              p            is convergent for all   p  &amp;gt;  1. This would lead one to conjecture something like:For all   ϵ  &amp;gt;  0, there exists an   N such that if   n  &amp;gt;  N, then there exists a prime between   n and   (  1  +  ϵ  )  n.Question: Is this conjecture true? If it is true, is there an expression for   N as a function of   ϵ?

odmeravan5c · Accepted Answer

That result follows from the Prime Number Theorem. You can make it effective with a result of Dusart: for   n  ≥  396738  , there is always a prime between   n and   n  +  n      /    (  25      log    2    ⁡  n  ). So in particular this holds for  N  ≥  max      (    exp    ⁡          (                        1                      25            ε                              )        ,         396738    )    .On the Riemann hypothesis (using the result of Schoenfeld) there is a prime between   x  −                    log        2            ⁡      x              x                    4      π       and   x for   x  ≥  599 and this should give a better bound.

Recall that Bertrand's postulate states that for n ≥ 2 there always exi

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