Related papers: Factorials and powers, a minimality result

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…

Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear…

A primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either…

In this paper we consider the equation $[p^{c}] + [m^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{29}{28}$, then it has a solution in a prime $p$ and an almost prime $m$ with at most $\left[…

Let P be a finite set of at least two prime numbers, and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A.…

In this paper, we consider the existence of a factorization of a monic, bounded motion polynomial. We prove existence of factorizations, possibly after multiplication with a real polynomial and provide algorithms for computing polynomial…

Factorization -- a simple form of standardization -- is concerned with reduction strategies, i.e. how a result is computed. We present a new technique for proving factorization theorems for compound rewriting systems in a modular way, which…

We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…

Factorial moments are convenient tools in particle physics to characterize the multiplicity distributions when phase-space resolution ($\Delta$) becomes small. They include all correlations within the system of particles and represent…

We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…

With reference to a baseline parametrization, we explore highly efficient fractional factorial designs for inference on the main effects and, perhaps, some interactions. Our tools include approximate theory together with certain carefully…

Let $k$ be a field. In this paper we will show that any factorial $\mathbb{A}^1$-form $A$ over any $k$-algebra $R$ is trivial, if $A$ has a retraction to $R$.

In this paper, we consider sums of values of degenerate falling factorials and give a probabilistic proof of a recurrence relation for them. This may be viewed as a degenerate version of the recent probabilistic proofs on sums of powers of…

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on…

This is the first one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple linear groups.

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in…

The Smarandache function of a positive integer $n$, denoted by $S(n)$, is defined to be the smallest positive integer $j$ such that $n$ divides the factorial $j!$. In this note, we prove that for any fixed number $k > 1$, the inequality…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally…