| In 1960, Sierpinski proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all positive integers n. Such integers are known as Sierpinski numbers. Letting f(x) = axr + bx + c ∈ Z [x], Chapter 2 of this document explores the existence of integers k such that f(k)2 n + d is composite for all positive integers n. Chapter 3 then looks into a polynomial variation of a similar question. In particular, Chapter 3 addresses the question, for what integers d does there exist a polynomial f(x) ∈ Z [x] with f(1) ≠ -d such that f(x)xn + d is reducible for all positive integers n. The last two chapters of the document then explore the reducibility and factorization of polynomials taking on a prescribed form. Specifically, Chapter 4 addresses the reducibility and factorization of polynomials of the form x n + cxn-1 + d ∈ Z [x], while Chapter 5 addresses the reducibility and factorization of polynomials of the more general form f( x)xn+ g( x) ∈ Z [x]. |
