### 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 - 2003 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

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

**Remarks on prime values of polynomials at prime arguments.** / Ruzsa, I.; Turjányi, Sándor.

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 62, no. 3-4, pp. 589-600.

}

TY - JOUR

T1 - Remarks on prime values of polynomials at prime arguments

AU - Ruzsa, I.

AU - Turjányi, Sándor

PY - 2003

Y1 - 2003

N2 - We show that for a large number N and for any functions taking integer values bounded by polynomials there exist integers t1, t2, ..., ts and primes 2 ≤ p1 2 <⋯ M ≤ N such that the numbers f1(pi) + t1, f2(pi) + t2, ..., fs(pi) + ts are primes for i = 1, 2, ..., M and M ≥ c N/ln Ns + 1 (where c is a positive constant independent of N), as expected. In Theorem 3.1 we show that when s = 1 and f1 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 f1 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.

AB - We show that for a large number N and for any functions taking integer values bounded by polynomials there exist integers t1, t2, ..., ts and primes 2 ≤ p1 2 <⋯ M ≤ N such that the numbers f1(pi) + t1, f2(pi) + t2, ..., fs(pi) + ts are primes for i = 1, 2, ..., M and M ≥ c N/ln Ns + 1 (where c is a positive constant independent of N), as expected. In Theorem 3.1 we show that when s = 1 and f1 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 f1 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.

UR - http://www.scopus.com/inward/record.url?scp=0037570736&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037570736&partnerID=8YFLogxK

M3 - Article

VL - 62

SP - 589

EP - 600

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 3-4

ER -