Narrow Results

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2ⁿ + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xⁿ + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽_p, and in that situation, we show 1 ∈ S for all primes p.

In this paper we investigate the factorization of trinomials of the form xⁿ + cx^n-1 + d ∈ ℤ[x]. We then use these results about trinomials to prove results about the factorization of polynomials of the form xⁿ + c(x^n-1 +⋯+ x + 1) ∈ ℤ[x].

Given relatively prime polynomials f(x) and g(x) in ℤ[x] with non-zero constant terms, we show that for n greater than an explicitly determined bound depending on f(x) and g(x), if the polynomial f(x)xⁿ + g(x) is non-reciprocal, then its non-cyclotomic part is irreducible except for some explicit cases where a known factorization of f(x)xⁿ + g(x) can easily be described. Prior work of a similar nature is discussed which shows under similar circumstances the non-reciprocal part off(x)xⁿ + g(x) is irreducible. The current paper establishes and makes use of a result which shows that a reciprocal polynomial f(x) with a root off the unit circle must have a root bounded away from the unit circle by an explicitly given function of the degree of f(x), the leading coefficient a of f(x) and the discriminant of f(x). Notably in this result, a need not be 1.

In 1908, Schur raised the question of the irreducibility over ℚ of polynomials of the form f(x) = (x - a₁)(x - a₂)⋯(x - a_n) + 1, where the a_i are distinct integers. Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the analogous question when replacing the linear polynomials with cyclotomic polynomials and allowing the constant perturbation of the product to be any integer d ∉ {-1, 0}. One interesting consequence of our investigations is that we are able to construct, for any positive integer N, an infinite set S of cyclotomic polynomials such that 1 plus the product of any k (not necessarily distinct) polynomials from S, where k ≢ 0(mod 2^N+1), is reducible over ℚ.