Joshua Harrington University of South Carolina On a Polynomial Variation of Sierpiński Numbers

In 1960 Sierpiński proved that there are infinitely many odd positive integers k such that k · 2 + 1 is composite for all positive integers n. A polynomial variation of Sierpiński’s result has been investigated by several people. More specifically, the question has been asked, for which integers d does there exist a polynomial f(x) ∈ Z[x] with f(1) 6= −d such that f(x) ·x + d is reducible over the rationals for all positive integers n. In 1967 Schinzel proved that there exists such a polynomial for d ≡ 0 (mod 12). This result was then extended in 2002 by Michael Filaseta who proved that such a polynomial exists for d ≡ 0 (mod 4). It was then shown in 2009 by Lenny Jones that one can find such a polynomial for infinitely many d ≡ 6 (mod 12). In this paper we further investigate this problem and improve upon previously known results.