Consecutive primes and IP sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909138279202816 |
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| author | Banks, William D. |
| author_facet | Banks, William D. |
| contents | For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each $n \ge 1$. A de Polignac number is any number m such that $p_{n+1}-p_n=m$ for infinitely many n. In this note, we show that every IP set of even natural numbers contains infinitely many de Polignac numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_10637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Consecutive primes and IP sets Banks, William D. Number Theory Combinatorics 11N05, 11N36, 11B75 For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each $n \ge 1$. A de Polignac number is any number m such that $p_{n+1}-p_n=m$ for infinitely many n. In this note, we show that every IP set of even natural numbers contains infinitely many de Polignac numbers. |
| title | Consecutive primes and IP sets |
| topic | Number Theory Combinatorics 11N05, 11N36, 11B75 |
| url | https://arxiv.org/abs/2403.10637 |