TITLE: At what point would an elementary generalization of Bertrand's Postulate be interesting?
QUESTION [13 upvotes]: I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting?  If an approach could show for example that for any $k$, there is a specific value $X$ which could be calculated such that for all $x \ge X$, there is a prime between $kx$ and $(k+1)x$, would this be interesting?
Or, does the Prime Number Theorem provide us enough insight that short of a proof of Legendre's Conjecture, elementary results are not very interesting at this time?

REPLY [2 votes]: Once again, being new here I tried to comment your answer, but obviously don't have 50 points yet.
Here is the original text of my comment with my mistake on value of x
"As a check on your equations I set k=100 x=2478 and should therefore expect one prime between 2478 and 2478 + 2478/100 = 2502.78.(for k=100, x must be greater than 2202.65 as detailed by your equations - this condition is met for x=2478).But there are no primes between 2478 and 2502.78. (Though 2503 is prime) Is the domain of x for your answer the positive integers? Then everything checks out okay. Or as you warned 'up to error'?"
Assigning x=24.78 is the correct value.
@quid: Note the prime is between kx and (k+1)x, so with my corrected value of x there should be a prime between 2478 < (some prime) < 2502.78, but the next prime is 2503.
@stefan kohl: understood and my calculation and answer/comment also gave x > 2202.65. My mistake was not putting the correct value for x