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 language English 589-600 12 Publicationes Mathematicae 62 3-4 Published - 2003

Pi
Polynomial
Integer
p.m.
Theorem
Exceed

ASJC Scopus subject areas

• Mathematics(all)

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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.
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