Abstract

Based on the data we get, nlog n is acceptable approximation to A(P_n) for small P_n ( most of relative errors are on the order of 1%, 1‰ or 0.1‰ for P_n less than 2000 ), where P_n is the n-th prime for n ≥ 2 and A(P_n) = L_n − P_n but L_n is the largest strong Goldbach number generated by P_n. Thus we propose a proposition that A(P_n) ≈ nlog n for all P_n. To give indirect verification of the proposition for large P_n, we obtain an experimental formula for calculating the number of primes not greater than P_n relying on existence of A(P_n). Using the formula, our found relative errors are generally smaller than that arising from x/log x, for example, there is a found relative error  to be about 0.00935% for the 382465573492-th prime but the relative error is about 3.45305% by x/log x, 0.16046% by x/((log x) − 1.08366), 0.02479% by x/log x + x/(log x)² + 2x/(log x)³. If the proposition is proven, then Goldbach’s conjecture is true.

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