Related papers: A Platonic basis of Integers
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with any irrational angle in degree: they are three $1$-parameter families of pentagonal subdivisions of the Platonic solids, with…
The reversal of a positive integer $A$ is the number obtained by reading $A$ backwards in its decimal representation. A pair $(A,B)$ of positive integers is said to be palindromic if the reversal of the product $A \times B$ is equal to the…
In this paper we describe a procedure for calculating the number of regular octahedrons that have vertices with coordinates in the set {0,1,...,n}. As a result, we introduce a new sequence in ``The Online Encyclopedia of Integer Sequences"…
The basis of the identity representation of a polyhedral group is able to describe functions with symmetries of a platonic solid, i.e., 3-D objects which geometrically obey the cubic symmetries. However, to describe the dynamic of assembles…
We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for $\lambda\in\{\frac{\pm1\pm\sqrt5}2,\pm\sqrt2,\pm\sqrt3\}$ that all integer sequences $(a_k)_{k\in\mathbb Z}$…
We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as…
In this article we consider numeric palindromes as a component of a pythagorean triple. We first show that there are infinitely many non-primitive pythagorean triples that contains (i) a single numeric palindrome as a component, (ii) two…
Polypolyhedra are edge-transitive compounds of polyhedra. In this paper we use group theory to determine the number of distinct polypolyhedra whose symmetry group is any given finite irreducible Coxeter group. We apply this result in order…
Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the…
A natural number is called an {\lambda}-parasitic number if it is multiplied by integer {\lambda} as the rightmost digit moves to the front. The Full set of these numbers is known in the decimal system. Here, a formula to analytically…
The satisfactory development of Quaternionic Analysis has indicated new solutions for physical and mathematical problems. It is worth mentioning the fact that quaternions possess four dimensions, and in this way they may be considered as…
A finite collection $P$ of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members of $P$. We associate with such a tiling a doubly infinite sequence with entries from $P$. The set of…
An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented…
We study the distribution of palindromic numbers (with respect to a fixed base $g\ge 2$) over certain congruence classes, and we derive a nontrivial upper bound for the number of prime palindromes $n\le x$ as $x\to\infty$. Our results show…
By the theory of elliptic curves, we study the integers representable as the product of the sum of four integers with the sum of their reciprocals and give a sufficient condition for the integers with a positive representation.
It is shown that Pythagorean triples can be used to generate matrices that have integer eigenvalues for all permutations of their coefficients, via simple formulas. For example, each and every permutation of the $2\times2$ matrix…
This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…
We introduce a notion of palindromicity of a natural number which is independent of the base. We study the existence and density of palindromic and multiple palindromic numbers, and we raise several related questions.
It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that…
To every partial order P, one associates a polynomial $\mathbb{D}_P$ in 4 variables that enumerates the intervals of P according to 4 parameters. Some symmetry properties of this polynomial are obtained for a specific family of posets, the…