Abstract

This paper is about the proof of P #x2260; NP based on the limiting factor that hinders the proof of P = NP when we consider randomness in the subset sum problem algorithm presented in this paper. Randomness in this paper is referred to an act of inputting integers to the program without pattern. The paper is not about to show case the extent in which the subset problem could be solved but to prove that P = NP does not exist in polynomial time. Literally, polynomial time means that as the complexity of the problem grows, the difficulty in solving it doesn#x2019;t grow too fast. The proof of x #xB1; y = b at system 1 is the premise that was used in this paper to prove that P #x2260; NP. The proof of x #xB1; y = b systems are forms of x #xB1; y = b that was derived from the coexistence of three quantities denoted by n, n + 1, n + 2 where n represents any positive integer. See the proof of x #xB1; y = b at references for details. The proof of x #xB1; y = b at system 1 is the proof of a mathematical method that proves something can evolve from nothing and its graph shows that the shape of the universe is a cone and this can further be mapped with an expanding universe or universes to locate the point of the big bang (a hypothetical point in space where the universe began). See No.4, at references for details.