Abstract

Using the first 4000000 primes to find L_n, the largest strong Goldbach number generated by the n-th prime P_n, we generalize a proposition in our previous work (Zhou 2017) and propose that L_n ≈ 2P_n and L_n/2P_n < 1 for sufficiently large P_n but the limit of L_n/2P_n as n → ∞ is 1, L_n ≈ P_n + n log n and L_n/(P_n + n log n) > 1 for sufficiently large P_n but the limit of L_n/(P_n + n log n) as n → ∞ is 1. There are five corollaries of the generalized proposition for getting L_n → ∞ as n → ∞, which is equivalent to Goldbach’s conjecture. If every step in distribution curve of L_n is called a Goldbach step, a study on the ratio of width to height for Goldbach steps supports the existence of above two limits but a study on distribution of Goldbach steps supports an estimation that Q(n) ≈ (1 + 1/log log n)n/log n and the limit of Q(n)/((1 + 1/log log n)n/log n) as n → ∞ is 1, where Q(n) is the number of Goldbach steps, from which we may expect there are infinitely many Goldbach steps to imply Goldbach’s conjecture.

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