Related papers: Pairing Pythagorean Pairs

A positive integer that is the area of some rational right triangle is called a congruent number. In an algebraic point of view, being a congruent number means satisfying a system of equations. As early as the 1800s, it is understood that…

We say that two elliptic curves E_1, E_2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character \chi of K, there is an isomorphism between the n-Selmer groups Sel_n(E_1^\chi/K) and…

A Thurston map $f\colon (S^2, A) \to (S^2, A)$ with marking set $A$ induces a pullback relation on isotopy classes of Jordan curves in $(S^2, A)$. If every curve lands in a finite list of possible curve classes after iterating this pullback…

We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group $(G,+)$ is a bijection $f:A\to B$ between two finite subsets $A,B$ of $G$ satisfying $a+f(a)\notin A$ for all $a\in A$. A…

Let $P$ be an arbitrary point on an elliptic curve over the complex numbers of the form $y^2=x^3+a_4\,x+a_6$ or of the form $y^2=x^3+a_2\,x^2+a_4\,x$. We provide explicit formulae to compute the points $P/2$, i.e., the points $Q$ such that…

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

Let E/Q be an elliptic curve, let L(E,s)=\sum a_n/n^s be the L-series of E/Q, and let P be a point in E(Q). An integer n > 2 having at least two distinct prime factors will be be called an elliptic pseudoprime for (E,P) if E has good…

For an integer $k\ge2$, a tuple of $k$ positive integers $(M_i)_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \[ \sigma(M_1)=\cdots=\sigma(M_k)=M_1+\cdots+M_k \] holds. This is a generalization of amicable pairs. An amicable…

A pair of numbers is amicable if each number equals the sum of the proper divisors of the other. This paper after exploring the history and evolution of amicable numbers, introduces a novel characterization of amicable pairs whose greatest…

We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries, which…

In the paper Algebraic quantum groups and duality I, we consider a pairing $(a,b)\mapsto\langle a,b\rangle$ of regular multiplier Hopf algebras $A$ and $B$. When $A$ has integrals and when $B$ is the dual of $A$, we can describe the duality…

Given a binary quadratic form $F \in \mathbb{Z}[X, Y]$, we define its value set $F(\mathbb{Z}^2)$ to be $\{F(x, y) : (x, y) \in \mathbb{Z}^2\}$. If $F$ and $G$ are two binary quadratic forms with integer coefficients, we give necessary and…

In this paper, we construct a family of elliptic curves with rank $\geq 5$. To do this, we use the Heron formula for a triple $(A^2, B^2, C^2)$ which are not necessarily the three sides of a triangle. It turns out that as parameters of a…

A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B),…

In this work, we define an orthogonal graph on the set of equivalence classes of $(2\nu + \delta)-$tuples over $\mathbb{Z}_{2^n}$ where $n$ and $\nu$ are positive integers and $\delta = 0, 1$ or $2$. We classify our graph if it is strongly…

Let $\mathcal{P}$ be a chiral polytope with type $\{k_1, k_2\}$ and $G=Aut(\mathcal{P})$. Suppose $|G|=2p^m$, where $k_1, k_2\geq 3$ and $p$ is an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes…

We characterize pairs of orthogonal countable ordinals. Two ordinals $\alpha$ and $\beta$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $\alpha$ and $\beta$ respectively such that the only…

Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is…

In this work, we identify a certain family of higher-dimensional formal groups over the ring of $p$-adic integers such that any two formal groups in that class coincide if they share infinitely many torsion points. As a useful application,…

We define and study a biadditive symmetric (not necessarily perfect) pairing on the torsion part $\mathrm{Pic}(X)_{\mathrm{tors}}$ of the Picard group of a smooth projective curve $X$ over a field $k$ with values in $k^\times \otimes…