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Polynomial cycles in certain local domains

T. Pezda — 1994

Acta Arithmetica

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple x , x , . . . , x k - 1 of distinct elements of R is called a cycle of f if f ( x i ) = x i + 1 for i=0,1,...,k-2 and f ( x k - 1 ) = x . The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number 7 7 · 2 N , depending only on the degree N of K. In this note we consider...

On prime values of reducible quadratic polynomials

W. NarkiewiczT. Pezda — 2002

Colloquium Mathematicae

It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer N r such that the polynomial f ( X ) / N r represents at least r distinct primes.

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