Related papers: Catalan's Equation For Even Primes over Quadratic …

Let $p$ and $q$ be different prime numbers. Using recent results of Ced\'o and Okni\'nski, we describe isomorphism classes of indecomposable set-theoretic solutions to the Yang--Baxter equation of cardinality $pq$.

We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific…

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes…

This paper deals with quadratic irrationals of the form $m/q+\sqrt v$ for fixed positive integers $v$ and $q$, $v$ not a square, and varying integers $m$, $(m,q)=1$. Two numbers $m/q+\sqrt v$, $n/q+\sqrt v$ of this kind are equivalent (in a…

Analysis of the dynamics of the Dyck words helped solve the problem of representing the Catalan number as a sum of squares of natural numbers. In this case, the Dyck triangle is considered in different coordinates. In the calculations, we…

In this paper, using a basic algebraic approach, we demonstrate that, if the integers $x,y$ and $z$ are pairwise relatively prime, and $x^p-my^p=zw$ where $m$ is an integer and $p$ is a prime, then we can write $x,y,m,z$ and $w$ as explicit…

We proved that any even number not less than 6 can be expressed as the sum of two old primes, $2n=p_i+p_j$

Two $(p,q)$-Laplace transforms are introduced and their relative properties are stated and proved. Applications are made to solve some $(p,q)$-linear difference equations.

Representations of primes by simple quadratic forms, such as $\pm a^2\pm qb^2$, is a subject that goes back to Fermat, Lagrange, Legendre, Euler, Gauss and many others. We are interested in a comprehensive list of such results, for $q\le…

Let $a,b,c $ be any positive integers such that $c\mid ab$ and $d_i^\pm$ is a square free positive integer of the form $d_i^\pm=a^{2k} b^{2l}\pm i c^m$ where $k,l \geq m$ and $i=1,2.$ The main focus of this paper to find the fundamental…

Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular,…

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…

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.

We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432d^3k^2$ over…

We verify the existence of radial positive solutions for the semi-linear equation $$ -\,\Delta u=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\geq 3$, $p$ is close to $p^*:=(N+2)/(N-2)$, and $V$ is a…

Shapiro proved an elegant convolution formula involving Catalan numbers of even index. This paper gives a simple combinatorial proof of his formula. In addition, we show that it is equivalent with the alternating convolution formula of…

We give two proofs of the $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{\vec{k}}(q,t)$ for $\vec{k}=(k_1,k_2,k_3)$. One is by using MacMahon's partition analysis as we proposed; the other is a direct bijection. We also prove…

We give a complete description of all solutions to the equation $f_1^3 + f_2^3 = f_3^3 + f_4^3$ for quadratic forms $f_j \in \mathbb C[x,y]$ and show how Ramanujan's example can be extended to three equal sums of pairs of cubes. We also…

The Super-Catalan numbers are a generalization of the Catalan numbers defined as $T(m,n) = \frac{(2m)!(2n)!}{2m!n!(m+n)!}$. It is an open problem to find a combinatorial interpretation for $T(m,n)$. We resolve this for $m=3,4$ using a…

There is a Paley graph for each prime power $q$ such that $q\equiv 1\pmod 4$. The vertex set is the field $\mathbb Fq$ and two vertices $x$ and $y$ are joined by an edge if and only if $x-y$ is a nonzero square of $\mathbb Fq$. We compute…