Is there a smallest real number a such that there exist a natural number N so that: n &gt; N ⟹ p n + 1 ≤ a ⋅ p n ?I believe it can be proved that n &gt; 7 ⟹ p n + 1 ≤ 2 ⋅ p n .

Question

Is there a smallest real number   a such that there exist a natural number   N so that:  n  &amp;gt;  N    ⟹        p          n      +      1        ≤  a  ⋅      p    n  ?I believe it can be proved that   n  &amp;gt;  7    ⟹        p          n      +      1        ≤      2    ⋅      p    n  .

Jazmyn Bean · Accepted Answer

There is no smallest such   a, but the infimum of the set of such   a is 1. In other words, for every   ϵ  &amp;gt;  0, there is a prime between   n and   (  1  +  ϵ  )  n for all sufficiently large   n. This follows from the prime number theorem.As a concrete example, for all   n  ≥  25, there is a prime between   n and       6    5    n

Gorlandint · Accepted Answer

Using the prime-number theorem:                                    p                      n            +            1                                      ≤        a        ⋅                  p          n                                                                              p                              n                +                1                                                    p                              n                                                                ≤        a                            (asymptotically:)                                                        (              n              +              1              )              log              ⁡              (              n              +              1              )                                      n              log              ⁡              (              n              )                                                  ≤        a                                                                      (              1              +              1                              /                            n              )                        1                                                              log              ⁡              (              n              +              1              )                                      log              ⁡              (              n              )                                                  ≤        a                                  (        1        +        1                  /                n        )        ⋅        (        1        +                                            log              ⁡              (              1              +              1                              /                            n              )                                      log              ⁡              (              n              )                                      )                    ≤        a            and the lhs on the last expression can be made arbitrarily near to 1 by increasing   n so also   a can be taken arbitrarily small by using appropriate   n

Is there a smallest real number a such that there exist a natural number N so that: n>N->p_(n+1)<=a x p_n?

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