avril 2016 – ideasfornumbertheory

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.