Generalized Sierpiński Numbers and New Irreducibility Criteria on Legendre Polynomials

A Sierpiński number is an odd positive integer k such that k·2n +1 is composite for all n ∈ Z+. Similarly, a Riesel number is an odd positive integer k such that k · 2n − 1 is composite for all n ∈ Z+, and a Brier number is a number that is both Sierpiński and Riesel. In this dissertation, we consider a generalization of a Sierpiński number and establish their existence by showing, for every fixed A ≥ 2, there are infinitely many odd positive integers k such that the k · an + 1 is composite for every integer a ∈ [2, A] and every n ∈ Z+. We produce the analogous Riesel-type result and Brier-type result. In an extension of the work of M. Filaseta, C. Finch-Smith, and M. Kozek in [16], we also show, for fixed integers R ≥ 1 and A ≥ 2, the existence of infinitely many odd positive integers k such that the expression kr · an + 1 is composite for every integer r ∈ [1, R], every integer a ∈ [2, A], and every n ∈ Z+. Each of these results in fact corresponds to an arithmetic sequence of such k. Further, these arithmetic sequences form a congruence class of the form L (mod M ) where L and M are relatively prime. As such, the work of J. Maynard and D. K. L. Shiu in [32] and [40] respectively allows us to extend these results to classifications of prime numbers. Legendre polynomials, where the nth Legendre polynomial, written Pn(x), is of degree n, are a classical family of polynomials that first arose in A. M. Legendre’s [27] study of the forces of attraction between spheroids. It has long been conjectured (see the 1890 letter [44] from Stieltjes to Hermite) that P̅n(x) := Pn(x) is irreducible for n ≥ 2 even and P̅n(x) := Pn(x)/x is irreducible for n ≥ 3 odd. Using Newton polygons, along with other techniques, we present an algorithm to determine if P̅n(x) is irreducible for a given n. We then use this algorithm to deduce the irreducibility of P̅n(x) for all n ≤ 10^7. Using similar ideas, we produce several new irreducibility criteria for P̅n(x) that build on prior results. Most notably, in an extension of existing irreducibility criteria, we show, for a fixed odd prime q, that P̅p+q(x) is irreducible for every prime p that is not in some finite exceptional computable set of primes Pq. We demonstrate the computational aspects of this result by showing P̅p+q(x) is irreducible for every odd prime q ≤ 13 and every prime p, and we obtain some additional partial results for q ≤ 29.