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Is 3-SAT $\leq_{p}$ Primality? And/or is Primality $\leq_{p}$ 3-SAT? I think the answer is no and yes, respectively, but I'm not sure. Any help would be appreciated.

Thank you.

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Probably you mean "Prime" with "Primality" (i.e. to know if a number is prime or not, which is in P :see the AKS article "Prime is in P").

Due to the NP completeness of the 3-SAT problem, I think your question:

  • is Primality ≤p 3-SAT

has TRUE like an answer, like you think. In fact every problem in NP can be reduced to 3-SAT in polynomial time, so every problem in P also, because every language in P is also in NP.

The other statement, 3-SAT ≤p Primality, would imply Primality to be NP-Complete and is not know for the moment. Anyway IF SO, P = NP because Primality is NP-Complete and Primality is in P.

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