Related papers: Diophantine equations, Platonic solids, McKay corr…

We search points in a Vogel plane with regular universal expression for character of adjoint representation. This gives seven patterns of singularities cancellation, each giving a certain Diophantine equation of third order on three…

We globally classify two-component evolution equations, with homogeneous diagonal linear part, admitting infinitely many approximate symmetries. Important ingredients are the symbolic calculus of Gel'fand and Dikii, the Skolem-Mahler-Lech…

Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-c})$. In this paper, we consider the Diophantine equation…

Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $\alpha$ with $\alpha>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n…

We completely solve the Diophantine equation $x^2+2^k11^\ell19^m=y^n$ in integers $x,y\geq 1;~ k,\ell, m\geq 0~$ and $n\geq 3$ with $\gcd(x,y)=1$, except the case $2\mid k, 2\nmid \ell m$ and $5\mid n$. We use this result to recover some…

Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component. They are also natural Poincar\'e duals to 1-cocycles with $\ZZ/2\ZZ$-coefficients. For a fixed cohomology…

In this paper we consider the Diophantine equation $x^2+q^{2m}=2y^p$ where $m,p,q,x,y$ are integer unknowns with $m>0,$ $p$ and $q$ are odd primes and $\gcd(x,y)=1.$ We prove that there are only finitely many solutions $(m,p,q,x,y)$ for…

Representation theory, for the classical binary polyhedral groups is encoded by the affine Dynkin diagrams E6^{(1)}, E7^{(1)} and E8^{(1)} (McKay correspondance). The quantum versions of these classical geometries are associated with…

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers…

Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the…

We study the $m$-graded quiver theories associated to CY $(m+2)$-folds and their order $(m+1)$ dualities. We investigate how monodromies give rise to mutation invariants, which in turn can be formulated as Diophantine equations…

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers…

The subject matter of this work is the diophantine equation x^n+y^m=c(x^k)(y^l), where n,m,k,l,c are natural numbers.We investigate this equation from the point of view of positive integer solutions.A preliminary examination of sources such…

The definition and properties of the Euler-Lagrange cohomology groups $H^{2k-1}$, $1 \leqslant k \leqslant n$, on a symplectic manifold $({\cal M}^{2n},\omega)$ are given and studied. For $k = 1$ and $k = n$, they are isomorphic to the…

Bidirected graphs are multigraphs where every edge has an independent direction at each end. In the paper, with an arbitrary bidirected graph we associate a non-negative integral quadratic form (called the incidence form of the graph), and…

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.

We consider the equality of the values of the $n$th and $k$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. For $k < n$, we prove that this equation always has a solution, but only finitely many…

In the fall 2011 issue of the Journal'Mathematics and Computer Education', author Unal Hasan, in the one page article "Proof without Words", gives a purely geometric proof of the equality, arctan(1/3)+ arctan(1/7) = arctan(1/2) (1) (See…

This paper considers Platonic solids/polytopes in the real Euclidean space R^n of dimension 3 <= n < infinity. The Platonic solids/polytopes are described together with their faces of dimensions 0 <= d <= n-1. Dual pairs of Platonic…

We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the…