A proof of the twin prime conjecture
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arXiv
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866914376463679488 |
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| author | Agama, Theophilus |
| author_facet | Agama, Theophilus |
| contents | In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed and $\mathbb{P}$ is the set of all prime numbers. In particular, it implies \begin{align}
\sum \limits_{p,p+2\in \mathbb{P}}1=\infty\nonumber \end{align} when we take $x\longrightarrow \infty$ on both sides of the inequality. We start by developing a general method for estimating correlations of the form \begin{align} \sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align} for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1707_03265 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A proof of the twin prime conjecture Agama, Theophilus General Mathematics 11Axx, 11Nxx In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed and $\mathbb{P}$ is the set of all prime numbers. In particular, it implies \begin{align} \sum \limits_{p,p+2\in \mathbb{P}}1=\infty\nonumber \end{align} when we take $x\longrightarrow \infty$ on both sides of the inequality. We start by developing a general method for estimating correlations of the form \begin{align} \sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align} for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$. |
| title | A proof of the twin prime conjecture |
| topic | General Mathematics 11Axx, 11Nxx |
| url | https://arxiv.org/abs/1707.03265 |