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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

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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.

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  • $\begingroup$ 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} $ ? $\endgroup$ – Ahmad Oct 7 '18 at 14:32
  • $\begingroup$ @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. $\endgroup$ – GH from MO Oct 7 '18 at 20:28

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