Related papers: Pythagorean Triples in the Fibonacci Model Set
From Euclid's fundamental formula for the Pythagorean triples we define the rational triples relating certain congruent numbers by an identity and explore their relationships. We introduce two geometric methods relating the congruent number…
In one of the three 2010/2011 issues of the journal 'MathematicalSpectrum', this author gave a three-parameter description of the entire set of integral triangles(i.e. triangles with integer side lengths)and with a 120 degree angle.This…
For a perfectoid ring $R$ and a natural number $n$ we investigate the essential image of the category of truncated by $n$ Barsotti-Tate groups under the anti-equivalence between commutative, finite, locally free, $R$-group schemes of…
A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$ \Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$…
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the…
For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent…
One shows that the Last Fermat Theorem is equivalent to the statement that all rational solutions of the famous equation are provided by an orbit of rationally parametrized subgroup of a group preserving k-ubic form. This very group…
We propose a new approach at Fermat's Last Theorem (FLT) solution: for each FLT equation we associate a polynomial of the same degree. The study of the roots of the polynomial allows us to investigate the FLT validity. This technique,…
We introduce a duality triads` notion. These are dual recurrences as used in dynamical data bases theory completed by a third pertinacious relation. Several representative examples of them are given. q-Gaussian triads as well as Fibonomial…
We provide a structure theorem for all almost complete intersection ideals of depth three in any Noetherian local ring. In particular, we find that the minimal generators are the pfaffians of suitable submatrices of an alternating matrix.…
Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on…
We study the ring of all functions from the positive integers to some field. This ring, which we call \emph{the ring of number-theoretic functions}, is an inverse limit of the ``truncations'' \Gamma_n consisting of all functions f for which…
Hierarchical sets such as the Pythagorean triplets ($\cal{PT}$) and the integral Apollonian gaskets ($\cal{IAG}$) are iconic mathematical sets made up of integers that resonate with a wide spectrum of inquisitive minds. Here we show that…
We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The representations considered are seen to be determined up to change of bases by a single number. Studying this number leads to a…
The Pythagorean triples have the structure of a ternary rooted tree; the tree is based on the Cayley graph of a free subgroup of the modular group
We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a…
We introduce a recursive theory that completely axiomatizes the structure $\langle \mathbb{Z},<, +,f,0\rangle$ where $f$ is the function that maps each $x$ to the integer part of $\varphi x $, with $\varphi$ the golden ratio. We prove that…
Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the resolution of certain Diophantine equations over $\mathbb{Q}$. Numerous novel obstacles arise when extending this approach to the resolution of Diophantine…
We prove a theorem which gives a bijection between the support $\tau$-tilting modules over a given finite-dimensional algebra $A$ and the support $\tau$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the…
This is an English translation of the author's Ph.D. thesis, accumulating his results on a construction of Cohen-Macaulay modules over a polynomial ring that appeared in the study of Cauchy-Fueter equations. This construction is generalized…