Polynomial Cunningham Chains

Let $\epsilon\in \{-1,1\}$. A sequence of prime numbers $p_1, p_2, p_3, ...$, such that $p_i=2p_{i-1}+\epsilon$ for all $i$, is called a {\it Cunningham chain} of the first or second kind, depending on whether $\epsilon =1$ or -1 respectively. If $k$ is the smallest positive integer such that $2p_k+\epsilon$ is composite, then we say the chain has length $k$. Although such chains are necessarily finite, it is conjectured that for every positive integer $k$, there are infinitely many Cunningham chains of length $k$. A sequence of polynomials $f_1(x), f_2(x), ...$, such that $f_i(x)\in \Z[x]$, $f_1(x)$ has positive leading coefficient, $f_i(x)$ is irreducible in \Q[x], and $f_i(x)=xf_{i-1}(x)+\epsilon$ for all $i$, is defined to be a {\it polynomial Cunningham chain} of the first or second kind, depending on whether $\epsilon =1$ or -1 respectively. If $k$ is the least positive integer such that $f_{k+1}(x)$ is reducible over \Q, then we say the chain has length $k$. In this article, for chains of each kind, we explicitly give infinitely many polynomials $f_1(x)$, such that $f_{k+1}(x)$ is the only term in the sequence $\{f_i(x)\}_{i=1}^{\infty}$ that is reducible. As a first corollary, we deduce that there exist infinitely many polynomial Cunningham chains of length $k$ of both kinds, and as a second corollary, we have that, unlike the situation in the integers, there exist infinitely many polynomial Cunningham chains of infinite length of both kinds

Multiplicative Bases for the Centres of the Group Algebra and Iwahori-Hecke Algebra of the Symmetric Group

Let \H_n be the Iwahori-Hecke algebra of the symmetric group $S_n$, and let Z(\H_n) denote its centre. Let $B={b_1,b_2,...,b_t}$ be a basis for Z(\H_n) over $R=\Z[q,q^{-1}]$. Then $B$ is called \emph{multiplicative} if, for every $i$ and $j$, there exists $k$ such that $b_ib_j= b_k$. In this article we prove that there are no multiplicative bases for $Z(\Z S_n)$ and Z(\H_n) when $n\ge 3$. In addition, we prove that there exist exactly two multiplicative bases for $Z(\Z S_2)$ and none for Z(\H_2).Comment: 6 pages. To appear in Proceedings of the Southeastern Lie Theory Workshop Series, Proceedings of Symposia in Pure Mathematic

On bases of centres of Iwahori-Hecke algebras of the symmetric group

Using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group over \Q[\xi] in 1990. An integral "minimal" basis was later given by the first author in 1999, following work of Geck and Rouquier. In principle one can then write elements of the norm basis as integral linear combinations of minimal basis elements. In this paper we find an explicit non-recursive expression for the coefficients appearing in these linear combinations. These coefficients are expressed in terms of readily computable numbers involving orders of symmetric groups and conjugacy classes. In the process of establishing this main theorem, we prove the following items of independent interest: a result on the projection of the norms onto parabolic subalgebras, the existence of an inner product on the Hecke algebra with some interesting properties, and the existence of a partial ordering on the norms.Comment: 29 pages. To appear J. Algebra. Original version January 200

Maximal Denumerant of a Numerical Semigroup With Embedding Dimension Less Than Four

Given a numerical semigroup $S =$ and $s\in S$, we consider the factorization $s = c_1 a_1 + c_2 a_2 +... + c_t a_t$ where $c_i\ge0$. Such a factorization is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such factorizations of $s$. We show that the number of maximal factorizations, varying over the elements in $S$, is always bounded. Thus, we define \dx(S) to be the maximum number of maximal factorizations of elements in $S$. We study maximal factorizations in depth when $S$ has embedding dimension less than four, and establish formulas for \dx(S) in this case.Comment: Main results are unchanged, but proofs and exposition have been improved. Some details have been changed considerably including the titl

A Connection Between the Monogenicity of Certain Power-Compositional Trinomials and kk-Wall-Sun-Sun Primes

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $$\{1,\theta,\theta^2,\ldots, \theta^{N-1}\}$$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Let $k$ be a positive integer, and let $U_n:=U_n(k,-1)$ be the Lucas sequence $\{U_n\}_{n\ge 0}$ of the first kind defined by U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}. A $k$-Wall-Sun-Sun prime is a prime $p$ such that $$U_{\pi_k(p)}\equiv 0 \pmod{p^2},$$ where $\pi_k(p)$ is the length of the period of $\{U_n\}_{n\ge 0}$ modulo $p$. Let ${\mathcal D}=k^2+4$ if $k\equiv 1 \pmod{2}$, and ${\mathcal D}=(k/2)^2+1$ if $k\equiv 0 \pmod{2}$. Suppose that $k\not \equiv 0 \pmod{4}$ and ${\mathcal D}$ is squarefree, and let $h$ denote the class number of ${\mathbb Q}(\sqrt{{\mathcal D}})$. Let $s\ge 1$ be an integer such that, for every odd prime divisor $p$ of $s$, ${\mathcal D}$ is not a square modulo $p$ and $\gcd(p,h{\mathcal D})=1$. In this article, we prove that $x^{2s^n}-kx^{s^n}-1$ is monogenic for all integers $n\ge 1$ if and only if no prime divisor of $s$ is a $k$-Wall-Sun-Sun prime

Characteristic Polynomials of Simple Ordinary Abelian Varieties over Finite Fields

We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${\mathcal A}$ of dimension $g$ over a finite field ${\mathbb F}_q$, when $q\ge 4$ and $2g=\rho^{b-1}(\rho-1)$ for some prime $\rho\ge 5$, with $b\ge 1$. Moreover, we show that ${\mathcal A}$ is absolutely simple if $b=1$ and $\rho=5$, but ${\mathcal A}$ is not absolutely simple for arbitrary $\rho$ with $b>1$