Is there always a prime between a prime and prime plus the index of that prime?

Question

Is it known is there always a prime strictly between $p_n$ and $p_n+n$, where $p_n$ is the $n$-th prime number and $n\geq5$?

I know about Bertrand`s postulate which states that for any integer $n>3$ there is always a prime $p$ such that $n<p<2n-2$.

If we would plug $p_n$ instead of $n$ we would get $p_n<p<2p_n-2$ but since I guess $p_n+n<2p_n-2$ will hold for all but finitely many $n$ we have that this problem of mine is stronger than Bertrand`s postulate and it seems that it is not implied by it.

So, is this known?

Answer 1

jagy@phobeusjunior:~$ 

     p    p + n  n    
     2     3     1    p/log p  2.885390081777927    p/log^2 p  4.162737962011215
 NONONONONO  3
     3     5     2    p/log p  2.730717679880512    p/log^2 p  2.485606349070669
 NONONONONO  5
     5     8     3    p/log p  3.106674672798059    p/log^2 p  1.930285504520985
     7    11     4    p/log p  3.597288396588255    p/log^2 p  1.848640544032643
 NONONONONO  11
    11    16     5    p/log p  4.587356305666709    p/log^2 p  1.913076170467284
    13    19     6    p/log p  5.06832618826664    p/log^2 p  1.975994642359189
    17    24     7    p/log p  6.00025410570094    p/log^2 p  2.117826431351823
    19    27     8    p/log p  6.452842166007064    p/log^2 p  2.191535369442038
    23    32     9    p/log p  7.335366744787423    p/log^2 p  2.339461099153619
    29    39    10    p/log p  8.612251926827733    p/log^2 p  2.557616663832689
    31    42    11    p/log p  9.027406962818835    p/log^2 p  2.628841176527419
    37    49    12    p/log p  10.24670205612772    p/log^2 p  2.837700081812219
    41    54    13    p/log p  11.04058283064003    p/log^2 p  2.973035835127401
    43    57    14    p/log p  11.43252118401864    p/log^2 p  3.039593967977556
    47    62    15    p/log p  12.20732420209676    p/log^2 p  3.170612003725475
    53    69    16    p/log p  13.34914438384008    p/log^2 p  3.362257656237909
    59    76    17    p/log p  14.46951764677999    p/log^2 p  3.548592219160638
    61    79    18    p/log p  14.83869415980414    p/log^2 p  3.609620399478779
    67    86    19    p/log p  15.93457740311697    p/log^2 p  3.789712791282478
    71    91    20    p/log p  16.65618860623435    p/log^2 p  3.907445336428887
    73    94    21    p/log p  17.01449483472118    p/log^2 p  3.965658006585665
    79   101    22    p/log p  18.08008761459324    p/log^2 p  4.137842634827439
    83   106    23    p/log p  18.7832074896648    p/log^2 p  4.250709440961442
    89   113    24    p/log p  19.82784807433869    p/log^2 p  4.417343362461308
    97   122    25    p/log p  21.20352530592732    p/log^2 p  4.634943148444331
     p  p + n    n    

From Dusart's results, we need check only for $p < 4000.$ In that range, we always get $p_{n+1} \leq p_n + n,$ with equality only at $p_{n+1} = 3,5,11.$

As you can see in the output, $n$ is about $p / \log p$ and much larger than $p / \log^2 p,$ even for fairly small numbers. Indeed, from Rosser and Schoenfeld (1962) we have, for $n \geq 6,$ $$ p_n > n \log n, $$ but $$ p_n < n \log n + n \log \log n $$

yes. First, this is reasonable, as $n \approx \frac{p}{\log p}.$ That is, you are asking, more or less, whether we have a prime between $p$ and $p + \frac{p}{\log p} $

We have the result of Dusart, page 8, theorem 6.8, that there is a prime between $x$ and $$ x + \frac{x}{25 \log^2 x} $$ as long as $x \geq 396738.$

Here we go, Pierre Dusart in his Ph. D. dissertation, for a lower bound gives a milder outcome that is good enough, there is prime between $x$ and $$ x + \frac{x}{2 \log^2 x} $$ as long as $x \geq 3275.$

BETTER BERTRAND