A proof of cases of de Polignac's conjecture
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866929437378871296 |
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| author | Mothebe, Mbakiso F. Kagiso, Dintle N. Modise, Ben T. |
| author_facet | Mothebe, Mbakiso F. Kagiso, Dintle N. Modise, Ben T. |
| contents | For $n \geq 1$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $$S= \{1,7,11,13,17,19,23,29 \},$$ the set of positive integers
which are both less than and relatively prime to $30.$ For $ x \geq 0,$ let \\ $T_x := \{ 30x+i \; | \; i \in S\}.$ For each $ x,$ $T_x$ contains at most seven primes.
Let $[ \; ]$ denote the floor or greatest integer function.
For each integer $s \geq 30$ let $π_7(s)$ denote
the number of integers $x, \; 0 \leq x < [\frac {s}{30}]$ for which $T_x$ contains seven primes. Let $m \geq 10^{10}$ be an integer and let $P_{K_m}$ denote the largest prime number less than $\sqrt{\prod_{i=1}^{m}p_i}.$
In this paper we show that $$\frac{\prod_{i=1}^{m}p_i}{8(K_m+1)} < π_7\left(\prod_{i=1}^{m}p_i\right) $$ and thereby prove that
there are infinitely
many values of $x$ for which $T_x$ contains seven primes. This, in particular, proves the well known twin prime conjecture as well as several cases of Alphonse de Polignac's conjecture that for every even number $k,$ there are infinitely many pairs of prime numbers
$p$ and $p'$ for which $p'-p = k.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_09290 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A proof of cases of de Polignac's conjecture Mothebe, Mbakiso F. Kagiso, Dintle N. Modise, Ben T. General Mathematics 11N05, 11N36 For $n \geq 1$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $$S= \{1,7,11,13,17,19,23,29 \},$$ the set of positive integers which are both less than and relatively prime to $30.$ For $ x \geq 0,$ let \\ $T_x := \{ 30x+i \; | \; i \in S\}.$ For each $ x,$ $T_x$ contains at most seven primes. Let $[ \; ]$ denote the floor or greatest integer function. For each integer $s \geq 30$ let $π_7(s)$ denote the number of integers $x, \; 0 \leq x < [\frac {s}{30}]$ for which $T_x$ contains seven primes. Let $m \geq 10^{10}$ be an integer and let $P_{K_m}$ denote the largest prime number less than $\sqrt{\prod_{i=1}^{m}p_i}.$ In this paper we show that $$\frac{\prod_{i=1}^{m}p_i}{8(K_m+1)} < π_7\left(\prod_{i=1}^{m}p_i\right) $$ and thereby prove that there are infinitely many values of $x$ for which $T_x$ contains seven primes. This, in particular, proves the well known twin prime conjecture as well as several cases of Alphonse de Polignac's conjecture that for every even number $k,$ there are infinitely many pairs of prime numbers $p$ and $p'$ for which $p'-p = k.$ |
| title | A proof of cases of de Polignac's conjecture |
| topic | General Mathematics 11N05, 11N36 |
| url | https://arxiv.org/abs/1912.09290 |