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2016 Polignac's Conjecture with New Prime Number Theorem
YinYue Sha
J. Phys. Math. 7(4): 1-5 (2016). DOI: 10.4172/2090-0902.1000201

## Abstract

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.

## Citation

YinYue Sha. "Polignac's Conjecture with New Prime Number Theorem." J. Phys. Math. 7 (4) 1 - 5, 2016. https://doi.org/10.4172/2090-0902.1000201

## Information

Published: 2016
First available in Project Euclid: 9 September 2017

Digital Object Identifier: 10.4172/2090-0902.1000201

Keywords: bilateral sieve method , Polignac prime , twin prime