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In this paper we consider two new conjectures concerning $D(4)$-quadruples and prove some special cases which support their validity. The main result is a proof that $\{a,b,c\}$ and $\{a+1,b,c\}$ cannot both be $D(4)$-triples.

In an earlier paper [4], we derived asymptotic formulas for the number of representations of zero and of large positive integers by the cubic forms in seven variables which can be written as $L_1(x_1,x_2,x_3) Q_1(x_1,x_2,x_3)+…

Let $b$ be a positive integer greater than 1, $N$ a positive integer relatively prime to $b$, $ |b|_{N}$ the order of $b$ in the multiplicative group $% \mathbb{U}_{N}$ of positive integers less than $N$ and relatively primes to $% N,$ and…

Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, G\'{o}mez, Gritschacher, Lind, and Tillman. In this article, we use unstable methods to construct explicit…

We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures.…

Let $0<k\in\mathbb{Z}$. We zipper-merge integer compositions with sums $k$ and $k+1$, equal number of parts and initial entries equal at least to 1 and 2, respectively. This yields bitstrings with two initial zeros, $k-1$ remaining zeros…

In this paper it is dealt with the following system of difference equations x_{n+1}=((a_{n})/(x_{n}))+((b_{n})/(y_{n})), y_{n+1}=((c_{n})/(x_{n}))+((d_{n})/(y_{n})), n in N_0, where the initial values x_0,y_0 are positive real numbers and…

Let $A$ and $B$ be sets of nonnegative integers. For a positive integer $n$ let $R_{A}(n)$ denote the number of representations of $n$ as the sum of two terms from $A$. Let $\displaystyle s_{A}(x) = \max_{n \le x}R_{A}(n)$ and…

Let $d$ be a fixed positive integer and let $\epsilon>0$. It is shown that for every sufficiently large $n\geq n_0(d,\epsilon)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side…

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted cubes $F(x_1,\ldots,x_s) = (x_1 - \mu_1)^3 + \ldots + (x_s - \mu_s)^3$. We show that if $\eta$ is real, $\tau >0$ is sufficiently large, and…

In this paper a closed form expression for the number of tilings of an $n\times n$ square border with $1\times 1$ and $2\times1$ cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to $m\times n$…

We count tilings of the $n \times m$ rectangular grid, cylinder, and torus with arbitrary tile sets up to arbitrary symmetries of the square and rectangle, along with cyclic shifting of rows and columns. This provides a unifying framework…

The effective dc-conductivity problem of isotropic, two-dimensional (2D), three-component, symmetric, regular composites is considered. A simple cubic equation with one free parameter for $\sigma_{e}(\sigma_1,\sigma_2,\sigma_3)$ is…

We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…

For k>=3 let A \subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A…

We compute the algebraic K-theory of the non-commutative ring k<x_1,...,x_n>/(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in…

Motivated by juggling sequences and bubble sort, we examine permutations on the set {1,2,...,n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive…

Given a periodic placement of copies of a tromino (either L or I), we prove co-RE-completeness (and hence undecidability) of deciding whether it can be completed to a plane tiling. By contrast, the problem becomes decidable if the initial…

Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}…