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We know that if we assume the Cramer's conjecture on prime gaps then given $k$ we can generate a prime $>2^k$ in deterministic time $O(k^c)$.

We also know that if we assume the Riemann's conjecture then given $k$ the best algorithm we have to generate a prime $>2^k$ is deterministic time $O(2^{\frac k2+\epsilon})$ (this is matched unconditionally by https://terrytao.files.wordpress.com/2011/02/polymath.pdf - from Ricky Demer).

What is the best algorithm we know assuming the ABC conjecture?

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    [This paper](https://terrytao.files.wordpress.com/2011/02/polymath.pdf#page=3) describes "a deterministic prime-finding algorithm that unconditionally takes $O\hspace{-0.04 in}\left(N^{1/2+o(1)}\hspace{-0.02 in}\right)$ time (thus matching the algorithm that was conditional on the Riemann hypothesis)". ​ ​ – 2017-01-03
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    @RickyDemer it looks like $\frac12-c+O(1)$ (theorem 1.2) (better than what you write). – 2017-01-03
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    ​ That theorem is only for "an _odd number_ of primes". ​ ​ ​ ​ – 2017-01-03
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    @RickyDemer $1$ is odd + Bertrand's postulate? – 2017-01-03
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    ​ 2 is even + the interval [4,8] ​ ​ ​ ​ – 2017-01-03
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    @RickyDemer thank you. – 2017-01-03

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