Remarks on prime values of polynomials at prime arguments

Imre Z. Ruzsa, Sándor Turjányi

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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 - Jan 1 2003

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ASJC Scopus subject areas

  • Mathematics(all)

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