Related papers: Primitive Divisors of some Lehmer-Pierce Sequences

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In…

Let $n\ge 2$ be an integer and let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power. Given $\mathbb F_q$-affine hyperplanes $\mathcal A_1, \ldots, \mathcal A_n$ of $\mathbb F_{q^n}$ in general position, we…

In this paper, we explore the use of path idempotents for the Hecke algebra of type $A$ at roots of unity. For $q$ a primitive $\ell$-th root of unity we obain a non-unital imbedding of (a quotient of) the group algebra of $S_m$ into (a…

We study the discriminants of the minimal polynomials $\mathcal{P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive integers $n\equiv11\pmod{24}$. The historical precedent for doing so comes from Gross and Zagier,…

I study group theory (Kleiss-Kuijf) relations between purely multi-quark primitive amplitudes at tree level, and prove that they reduce the number of independent primitives to (n-2)!/(n/2)!, where n is the number of quarks plus antiquarks,…

In this article, we establish a sufficient condition for the existence of a primitive element $\alpha \in {\mathbb{F}_{q^n}}$ such that the element $\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and…

While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…

For integer a let us consider the sequence X_a={x_0,x_1,x_2,...} defined by x_0=a, x_1=1 and, for n>=1, x_{n+1}=x_n+x_{n-1}. We say that a prime p divides X_a if p divides at least one term of the sequence. It is easy to see that every…

We study primitive elements in the Ringel-Hall algebra H(A) of an algebra A over a finite field associated with a quiver with automorphism. When A is a tame hereditary algebra, we give a description of primitive elements in H(A) which…

Let $q$ be an odd prime power and write \[ \theta_q := \frac{\phi(q-1)}{q-1}. \] If $\theta_q < \tfrac{1}{3}$, or if $\theta_q = \tfrac{1}{3}$ and $q \notin \{7,13,19,25,37\}$, then the finite field $\F$ contains a pair of consecutive…

Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference…

We develop the ratios conjecture with one shift in the numerator and denominator in certain ranges for families of primitive quadratic Hecke $L$-functions of imaginary quadratic number fields with class number one using multiple Dirichlet…

Let $\omega^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of…

We investigate a class of algebras that provides multiparameter versions of both quantum symplectic space and quantum Euclidean $2n$-space. These algebras encompass the graded quantized Weyl algebras, the quantized Heisenberg space, and a…

In this paper, we first investigate the relationship between the number of primitive representations of $n$ by quadratic forms and the number of non-primitive ones. We hence obtain a theorem to deal with the Eisenstein series part with…

The primitive equations in a 3D infinite layer domain are considered with linearly growing initial data in the horizontal direction, which illustrates the global atmospheric rotating or straining flows. On the boundaries, Dirichlet, Neumann…

We define a group stucture on the primitive integer points (A,B,C) of the algebraic variety Q_0(B,C)=A^n, where Q_0 is the principal binary quadratic form of fundamental discriminant \Delta and n is a fixed integer greater than 1. A…

In this paper, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main…

We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must…