Expert
Analyzing the P vs NP Problem
Discuss the implications of P equaling NP versus P not equaling NP in the context of computational complexity.
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Provide a comprehensive theoretical analysis of the P vs NP problem. Explain the definitions of P and NP complexity classes. Discuss the theoretical consequences on cryptography, optimization, and mathematical proof verification if P were to equal NP. Conversely, explain the significance if they are proven to be unequal.