Related papers: The generalized Catalan equation in positive chara…

A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial…

Let $\mathbb{F}_q$ be a finite field of $q=p^k$ elements. For any $z\in \mathbb{F}_q$, let $A_n(z)$ and $B_n(z)$ denote the number of solutions of the equations $x_1^3+x_2^3+\cdots+x_n^3=z$ and $x_1^3+x_2^3+\cdots+x_n^3+zx_{n+1}^3=0$…

Let $k$ be a fixed positive integer with $k>1$. In this paper, using various elementary methods in number theory, we give criteria under which the equation $x^2+(2k-1)^y=k^z$ has no positive integer solutions $(x,y,z)$ with $y\in\{3,5\}$.

Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique.…

We extend the notion of polynomial integration over an arbitrary circle $C$ in the Euclidean geometry over general fields $\mathbb F$ of characteristic zero as a normalized $\mathbb F$-linear functional on $\mathbb{F}\left[\alpha_1,…

Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ with $x_i\in\mathbb{F}_{p^{t_i}}$, where $b\in\mathbb{F}_q$ and…

Fix non-zero reals $\alpha_1,\ldots,\alpha_n$ with $n\ge 2$ and let $K$ be a non-empty open connected set in a topological vector space such that $\sum_{i\le n}\alpha_iK\subseteq K$ (which holds, in particular, if $K$ is an open convex cone…

Let A be an arbitrary integral domain of characteristic 0 which is finitely generated over Z. We consider Thue equations $F(x,y)=b$ with unknowns x,y from A and hyper- and superelliptic equations $f(x)=by^m$ with unknowns from A, where the…

A linear configuration is said to be common in a finite Abelian group $G$ if for every 2-coloring of $G$ the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf…

Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study…

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…

We study the properties of the odd Catalan numbers, C_n, modulo 2^k for k >= 2. We show that there exist only k - 1 different congruences of the odd Catalan numbers modulo 2^k. Moreover, these congruences can be obtained by C_{2^m - 1} (mod…

For $ k \geq 2 $, we let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) =1$ and, for a non-negative integer $s$, the generalized Frobenius number of $A$, $g(A;s) = g(a_1, a_2,…

The Catalan transform of a sequence (a_{n})_{n>=0} is the sequence (b_{n})_{n>=0} with b_{n} = Sum[k/(2n-k) (2n-k)-choose-(n-k) a_{k},k=0..n]. Here we show that the Catalan transform of the Catalan numbers has a simple interpretation: it…

We consider the equation $P(Q(x_1,\ldots,x_\nu))=Q(P(x_1),\ldots,P(x_\nu))$ in polynomials over the field of complex numbers and prove that if ${\rm deg}(P)>1$, then it is only solvable in polynomials that are affinely conjugate to…

We extend the polynomial Pell's equation satisfied by univariate Chebyshev polynomials on [--1, 1] from one variable to several variables, using orthogonal polynomials on regular domains that include cubes, balls, and simplexes of arbitrary…

We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y+1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.

Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this…

We define the notion of a Catalan pair (which is a pair of binary relations (S,R) satisfying certain axioms) with the aim of giving a common language to most of the combinatorial interpretations of Catalan numbers. We show, in particular,…

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…