P=NP implies a deterministic algorithm to find primes

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Consider the decision problem of determining whether there is a prime in the range [m,n], where m < n. This problem is in NP, since if there is a prime in the range, we can efficiently verify that it is prime. Therefore, if P=NP, then this decision problem is in P, i.e., there is a polynomial-time algorithm to determine whether there is a prime in the range [m,n].

We can use this algorithm as a subroutine to quickly find a prime in the range [n,2n] for large n. We use the well-known fact that such a prime always exists. We then use the polynomial-time algorithm described above to do a binary search to locate such a prime. This terminates after at most log2n applications of the subroutine, and thus is a polynomial-time algorithm to generate a prime.

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