### Abstract

We show that for a large number N and for any functions taking integer values bounded by polynomials there exist integers t_{1}, t_{2}, ..., t_{s} and primes 2 ≤ p_{1} < p_{2} < ⋯ < p_{M} ≤ N such that the numbers f_{1}(p_{i}) + t_{1}, f_{2}(p_{i}) + t_{2}, ..., f_{s}(p_{i}) + t_{s} are primes for i = 1, 2, ..., M and M ≥ c N/ln N^{s + 1} (where c is a positive constant independent of N), as expected. In Theorem 3.1 we show that when s = 1 and f_{1} is a polynomial taking integer values the order of the magnitude of M can be improved by a factor log log N. Theorem 4.1 illustrates that there exists a function f_{1} which is "not far" from being a polynomial but the value of M never exceeds the expected order of magnitude, so in the improvement it is essential that the function is actually a polynomial.

Original language | English |
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Pages (from-to) | 589-600 |

Number of pages | 12 |

Journal | Publicationes Mathematicae |

Volume | 62 |

Issue number | 3-4 |

Publication status | Published - Jan 1 2003 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*62*(3-4), 589-600.