Effective short intervals containing primes

95 years ago Hoheisel proved the existence of primes in the sub-linear interval $$[x,x+x^{1-\frac{1}{33000}}]\text{for }x\text{ sufficiently large}\mathrm{.}$$ This was improved by Heilbronn, proving existence of primes in the sub-linear interval $$[x,x+x^{1-\frac{1}{250}}]\text{for }x\text{ sufficiently large}\mathrm{.}$$ More recently Baker, Harman, Pintz proved existence of primes in the sub-linear interval $$[x,x+x^{1-\frac{19}{40}}]\text{for }x\text{ sufficiently large}\mathrm{.}$$ In the present article I will, to the extent possible, make some of these statements effective. Specifically, among other things, I shall show that $$\mathrm{\forall }n\ge 4,\mathrm{\forall }x\ge \exp (3\exp (33)),\text{there are primes in the interval}[x,x+x^{1-\frac{1}{n}}];$$ $$\mathrm{\forall }n\ge 91,\mathrm{\forall }x\ge [{90}^{90}{]}^{n\mathrm{/}(n-90)},\text{there are primes in the interval}[x,x+x^{1-\frac{1}{n}}]\mathrm{.}$$ Furthermore $$\mathrm{\forall }n\ge 106,\mathrm{\forall }x\ge 1,\text{there are primes in the interval}[x,x+x^{1-\frac{1}{n}}]\mathrm{.}$$ In particular this last observation makes both the Hoheisel and Heilbronn results fully explicit and effective. This (relatively) specific observation can be extended and generalized in various manners.