P vs NP Problem
More realistically, “P=NP” is a Millennium Prize problem. So, if it is solved, then somebody gets a $1 million prize from the Clay Mathematics Institute.
Origin of NP-Completeness Theory
The theory of NP-completeness is typically traced back to Steve Cook’s 1971 paper “The complexity of theorem-proving procedures” [14], which provided the first published NP-completeness results.
Understanding P vs NP Question
Put simply, the P versus NP question asks whether the set of problems that can be easily solved are also in the set of problems that can be easily checked.
P and NP Class Definition
Step 1: If a problem is in class P, it is nothing but we can find a solution to that type of problem in polynomial time.
Step 2: If a problem is in class NP, it is nothing but that we can verify a possible solution in polynomial time.
Integer Factorization Complexity
Integer factorization is not NP-hard (so not NP-complete). (This isn’t proven, but it’s generally thought to be the case.)
Prime Factorization Completeness
No, it’s not known to be NP-complete, and it would be very surprising if it were. This is because its decision version is known to be in NP∩co-NP. (Decision version: Does n have a prime factor)