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.


For the first question: This is an open problem. If $\mathsf{P} \neq \mathsf{NP}$ then the answer is no: the decision version of 3-SAT is $\mathsf{NP}$-Complete, while Primality is in $\mathsf{P}$, the means that a Karp reduction from 3-SAT to Primality would imply $\mathsf{P}=\mathsf{NP}$.

For the second question: Primality is in $\mathsf{P} \subseteq \mathsf{NP}$, therefore there is a Karp reduction from Primality to any $\mathsf{NP}$-Complete problem, such as 3-SAT.

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