Abstract

Let L_n denote the largest strong Goldbach number generated by the n-th prime P_n, in this paper, we present that there are approximate bounds of L_n such that 2nlogn + 2nloglogn – 2n < L_n < 2nlogn + 2nloglogn for n ≥ 20542, based on results about prime number theorem. Let ξ(n) = L_n/2 denote the number of strong Goldbach numbers generated by P_n, equivalently, there are approximate bounds of ξ(n) such that nlogn + nloglogn – n < ξ(n) < nlogn + nloglogn for n ≥ 20542. The approximate bounds of L_n have been verified for 20542 ≤ n ≤ 400000000, equivalently, the approximate bounds of ξ(n) have also been verified for 20542 ≤ n ≤ 400000000. It is obvious that if it can be proven that there is an integer k > 0 such that bounds of 2P_n can be thought as approximate bounds of L_n for all n > k, or equivalently, there is an integer k > 0 such that bounds of P_n can be thought as approximate bounds of ξ(n) for all n > k, then Goldbach conjecture is true. We also considered another approach to the conjecture, that is, if it can be proven by introducing Li(n) that there are infinitely many Goldbach steps, then Goldbach conjecture is true.

This work is licensed under a Creative Commons Attribution 4.0 License.