On the Connections between Goldbach Conjecture and Prime Number Theorem

By our suggested definition, an even number L_n is called the largest strong Goldbach number generated by the n-th prime P_n if every even number from 4 to L_n is the sum of two primes not greater than P_n but L_n + 2 is not such a sum. We discovered the existence of step-type distribution for L_n arising from observed fact that L_n ≤ L_n+1 and we proved that L_n ≤ L_n+1 for all n > 0. Every such step is called a Goldbach step whose width is (n2 + 1) – n1, where n1 is the starting point and n2 is the finishing point for the step. We proved that if Goldbach conjecture is true then there are infinitely many Goldbach steps. It is expected that distribution of Goldbach steps is asymptotically expressed as Q(n) ~ n/logn same as prime number theorem, where Q(n) is the number of Goldbach steps. It means Goldbach steps have like-prime nature, thus, all n1 can be called like-primes and g_i = n1-(i + 1) – n1-i is defined as gap between the i-th and the (i + 1)-th like-primes. We proved that if there are infinitely many like-prime gaps whose length k is uncertain but bounded by a finite integer N > 1, then Goldbach conjecture is true. Considering k = 1 for twin like-primes, it is conjectured that there are infinitely many like-primes n1 such that n1 + 1 is also like-prime to imply Goldbach conjecture and it is expected that distribution of twin like-primes is asymptotically expressed as Q₂(n) ~ 2C₂n/(logn)² akin to prime number theorem and same as a special case of the first Hardy-Littlewood conjecture, where Q₂(n) is the number of twin like-primes and C₂ is twin prime constant. We also studied distributions of triplet like-primes and quadruplet like-primes to imply Goldbach conjecture. We presented there are bounds of L_n/2 such that nlogn + nloglogn – n < L_n/2 < nlogn + nloglogn for n ≥ 20542, and in this paper, the bounds have been verified up