# consecutive prime gaps and explicit bound

I am aware of the theorem that $$p_{n+1}- p_n \leq n^{0.525}$$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" explicit, is it known that $$p_{n+1}-p_n \leq c n^{\alpha}$$ for all $$n \geq 1$$ and for small $$c$$, lets say $$c \leq 2$$ and $$\alpha \leq 0.55$$ ?

Any ref that can give me the explicit numbers or a way to construct them would be great.

Thank you, also i posted the question yesterday on MSE

The result you quote is due to Baker-Harman-Pintz (2000). I am not aware of any concrete effective version of this result, but if you increase the exponent $$0.525$$ to $$2/3$$, then such a variant is available by the work of Dudek. See also my response to this MO question.
• Thx, so is there $\alpha <1$ such that $c\leq 2$ and $n_0$ or $x_0$ is small say less than $10^{10}$ or $10^{12}$ ? – Ahmad Oct 7 '18 at 14:32
• @Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_{n+1}-p_n<n^{3/4}$. It is not clear to me what can be said for smaller $n$'s unconditionally. – GH from MO Oct 7 '18 at 20:28