TITLE:
On the Connections between Goldbach Conjecture and Prime Number Theorem
AUTHORS:
Pingyuan Zhou
KEYWORDS:
Largest Strong Goldbach Number, Goldbach Step, Like-Prime, Polignac’s Conjecture, The First Hardy-Littlewood Conjecture, Like-Prime Gap, Twin Like-Prime Conjecture, Triplet and Quadruplet Like-Primes, Bounds of the Largest Strong Goldbach Number, Goldbach Conjecture, Prime Number Theorem
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.6,
June
26,
2025
ABSTRACT: By our suggested definition, an even number Ln is called the largest strong Goldbach number generated by the n-th prime Pn if every even number from 4 to Ln is the sum of two primes not greater than Pn but Ln + 2 is not such a sum. We discovered the existence of step-type distribution for Ln arising from observed fact that Ln ≤ Ln+1 and we proved that Ln ≤ Ln+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 gi = 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 Ln/2 such that nlogn + nloglogn – n Ln/2 nlogn + nloglogn for n ≥ 20542, and in this paper, the bounds have been verified up to n = 4000000000. If it can be proven that bounds of prime, nlogn + nloglogn – n Pn nlogn + nloglogn for n ≥ 6, can be used as bounds of Ln/2 for n ≥ 20542, then Goldbach conjecture is true. Further, we proved that if there is a bounded integer k > 20541 such that bounds of prime can be used as bounds of Ln/2 for n ≥ k then Goldbach conjecture is true, where bounded integer k > 20541 means value of k is uncertain but there exists upper bound N > 20542 for k.