Remarks on prime values of polynomials at prime arguments

I. Ruzsa, Sándor Turjányi

Research output: Contribution to journalArticle

Abstract

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 <p2 <⋯ <pM ≤ 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.

Original languageEnglish
Pages (from-to)589-600
Number of pages12
JournalPublicationes Mathematicae
Volume62
Issue number3-4
Publication statusPublished - 2003

Fingerprint

Pi
Polynomial
Integer
p.m.
Theorem
Exceed

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Publicationes Mathematicae, Vol. 62, No. 3-4, 2003, p. 589-600.

Research output: Contribution to journalArticle

Ruzsa, I. ; Turjányi, Sándor. / Remarks on prime values of polynomials at prime arguments. In: Publicationes Mathematicae. 2003 ; Vol. 62, No. 3-4. pp. 589-600.
@article{c8324f2c34b54dd08a8c4d1ee3b3f4d1,
title = "Remarks on prime values of polynomials at prime arguments",
abstract = "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.",
author = "I. Ruzsa and S{\'a}ndor Turj{\'a}nyi",
year = "2003",
language = "English",
volume = "62",
pages = "589--600",
journal = "Publicationes Mathematicae",
issn = "0033-3883",
publisher = "Kossuth Lajos Tudomanyegyetem",
number = "3-4",

}

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 -