Cramer’s conjecture implies twin prime conjecture

Let’s define the quantity G(x):=\sup\{p_{n+1}-p_{n}\mid \frac{p_{n}+p_{n+1}}{2}\le x\}, and assume Cramer’s conjecture. Then K_{m}:=\sup\{\frac{G(x)}{\log^2 x}\mid x>2\}<\infty. Let’s denote by U(n) the event n is 1-central, and by H(n) the event (n-1,n+1) is a couple of twin primes. Then from de Bayes’ theorem, one gets P(H(n))=\frac{P(H(n)\cap U(n))}{P(U(n)/H(n))}=P(H(n)\cap U(n)), since if n is the half sum of a couple of twin primes, then necessarily n is 1-central. But P(H(n)\cap U(n)) is equal to the ratio of the number of quantities 2r_{0}(n) such that n is the half sum of twin primes to the number G(n) as defined above, hence P(H(n))\ge\frac{1}{K_{m}\log^{2} n}. As P(H(n))=\frac{1}{2}\frac{\pi_{2}(n)}{n}, one finally gets \pi_{2}(x)\ge\frac{2x}{K_{m}\log^{2} x}. Hence the assumption of Cramer’s conjecture implies that there are infinitely many twin primes.

 

 

 

 

 

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