I know that in 1952 [Jitsuro Nagura](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195570997) 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?