Physical Mathematics

Polignac's Conjecture with New Prime Number Theorem

Abstract

YinYue Sha

There are infinitely many pairs of consecutive primes which differ by even number En.Let Po(N, En) be the number of Polignac Prime Pairs (which difference by the even integer En) less than an integer (N+En), Pei be taken over the odd prime divisors of the even integer En less than √(N+En), Pni be taken over the odd primes less than √(N+En) except Pei, Pi be taken over the odd primes less than √(N+En), then exists the formulas as follows:

Po(N, En) ≥ INT {N × (1-1/2) × Π (1-1/Pei) × Π (1-2/Pni)} - 1

≥ INT {Ctwin × Ke(N) × 2N/(Ln (N+En))^2} - 1

Po(N, 2) ≥ INT {0.660 × 1.000 × 2N/(Ln (N+2))^2} - 1

Π (Pi(Pi-2)/(Pi-1)^2) ≥ Ctwin=0.6601618158…

Ke(N)=Π( (1-1/Pei)/(1-2/Pei))=Π( (Pei-1)/(Pei-2)) ≥ 1

where -1 is except the natural integer 1.

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