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



consecutive prime gaps and explicit bound
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](https://arxiv.org/abs/1401.4233). See also my response to [this MO question](https://mathoverflow.net/questions/212816/are-there-effective-small-intervals-in-which-primes-are-dense).


