(242j) Structural Similarity Between the Ordered Pairs of Primes and Non-Primes Via the Process Systems Engineering Approach
Though a number of studies have been conducted on prime number, many parts of primes remain unknown. Riemann hypothesis, twin prime problem and Goldbach’s conjecture are typical examples of the famous unsolved problems. In 1742, Christian Goldbach suggested that every integer greater than 6 can be represented as a sum of three primes. This conjecture can be divided into two different statements: weak Goldbach’s conjecture and strong Goldbach’s conjecture. The weak Goldbach’s conjecture states that every integer greater than 5 can be written as the sum of three primes. Vinogradov showed that this conjecture holds for numbers larger than 3315 and this value is reduced to 3.33 × 1043000 by Chen. The strong Goldbach’s conjecture states that every even integer greater than 2 can be written as the sum of two primes. Unlike the weak conjecture, this conjecture still remains unsolved. There have been various approaches to prove Goldbach’s conjecture using analytical number theory. We go back to the starting point of this famous problem and are able to show that the number of Goldbach partition is related to that of ordered pairs of non-primes. It is based on the world’s first dynamic model of primes where the concept of process systems engineering is applied and can be a key to identify the structure of prime numbers. In this study, the structural similarity between the ordered pairs is shown as the first step for proving the strong Goldbach’s conjecture. We have shown that there is a structural similarity between the ordered pairs in Gn. Goldbach partition, n(an), and the number of expression for even number as the sum of the two composite number, n(dn) have a similar structure which can be desribed as Dn(an) = Dn(dn)+η(n). From this, it is possible to explain the similar tendency in the graph of both n(an) and n(dn) quantitatively. Meanwhile, it can be also found that the number of Goldbach partition has a special periodicity based on the suggested dynamic model.
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