Let $p_1,p_2,\cdots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k\ge 5$, then $$(p_{k+1}-1)!\mid \left(\frac{1}{2}(p_{k+1}-1)\right)!p_k!,$$ which improves a previous result of the second author.

The functional-differential equation $f^{\text{'}}\left(t\right)=1/f\left(f\left(t\right)\right)$ is closely related to Golomb’s self-described sequence $F$,$$\underset{1,}{\underbrace{1,}}\underset{2,}{\underbrace{2,2,}}\underset{2,}{\underbrace{3,3,}}\underset{3,}{\underbrace{4,4,4}}\underset{3,}{\underbrace{5,5,5,}}\underset{4,}{\underbrace{6,6,6,6,}}\cdots .$$We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number $p\ge 0$ there is exactly one increasing solution with $p$ as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely many times. In fact...

Some aspects of duality triads introduced recently are discussed. In particular, the general solution for the triad polynomials is given. Furthermore, a generalization of the notion of duality triad is proposed and some simple properties of these generalized duality triads are derived.

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

The Pell sequence ${\left(P_n\right)}_{n=0}^{\infty }$ is the second order linear recurrence defined by $P_n=2P_{n-1}+P_{n-2}$ with initial conditions $P_0=0$ and $P_1=1$. In this paper, we investigate a generalization of the Pell sequence called the $k$-generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers are also deduced....

In 1988 the first author and J. A. Guthrie published a theorem which characterizes the topological structure of the set of subsums of an infinite series. In 1998, while attempting to generalize this result, the second author noticed the proof of the original theorem was not complete and perhaps not correct. The present paper presents a complete and correct proof of this theorem.