Related papers: Tabulating Absolute Lucas Pseudoprimes

Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number…

Lucas' theorem describes how to reduce a binomial coefficient $\binom{a}{b}$ modulo $p$ by breaking off the least significant digits of $a$ and $b$ in base $p$. We characterize the pairs of these digits for which Lucas' theorem holds modulo…

We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to…

Let $U$ be a Lucas sequence, $p$ be prime, and $\rho_U(p)$ be the rank of appearance of $p$ in $U$. We derive closed-form formulas for the Dirichlet density of primes $p$ for which $d\mid \rho_U(p)$, where $d\geq 1$ is a fixed integer. Our…

This paper considers Weyl modules for a simple, simply connected algebraic group over an algebraically closed field $k$ of positive characteristic $p\not=2$. The main result proves, if $p\geq 2h-2$ (where $h$ is the Coxeter number) and if…

We establish that every set of $k=10$ natural numbers determines at least $30$ distinct pairwise sums or at least $30$ distinct pairwise products, as well as the analogous result for $k=11$ and at least $34$ sums/products, with sharpness…

Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes.…

Let $Q$ be a set of primes with relative density $\delta$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $\delta \in (0,1]$. This generalizes the…

We give estimates on the number $AL_H(x)$ of arithmetic lattices $\Gamma$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$.…

A composite positive integer n is Lehmer if \phi(n) divides n-1, where \phi(n) is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author [7] proved that there is no…

Let $X_{n}$ denote the unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$. We continue the study of cases in which the inequality $\gamma_t(X_n) \le g(n)$ is strict, where $\gamma_t$ denotes the total domination number, and $g$ is the…

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such…

This contribution presents all possible solutions to the Diophantine equations $F_k=L_mL_n$ and $L_k=F_mF_n$. To be clear, Fibonacci numbers that are the product of two arbitrary Lucas numbers and Lucas numbers that are the product of two…

To study finite-dimensional modules of the Lie superalgebras, Kac introduced the Kac-modules and divided them into typical or atypical modules according as they are simple or not. For Lambda being atypical, Hughes et al have an algorithm to…

We show that lambda calculus is a computation model which can step by step simulate any sequential deterministic algorithm for any computable function over integers or words or any datatype. More formally, given an algorithm above a family…

Let $(u_n)_{n \ge 0}$ be a nondegenerate Lucas sequence and $g_u(n)$ be the arithmetic function defined by $\gcd(n, u_n).$ Recent studies have investigated the distributional characteristics of $g_u$. Numerous results have been proven based…

In this paper, we extend the $p$-adic valuations originally obtained by Carmichael for the sequences obtained by applying M\"obius inversion to Lucas sequences to $p$-adic congruences, from which we immediately derive corresponding…

Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $\Delta(f_n)$ denote the discriminant of $f_n$, that is $\Delta(f_n) =…

Automatic differentiation is everywhere, but there exists only minimal documentation of how it works in complex arithmetic beyond stating "derivatives in $\mathbb{C}^d$" $\cong$ "derivatives in $\mathbb{R}^{2d}$" and, at best, shallow…

We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell…