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

Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.

Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are…

In this paper we consider the generalized Catalan numbers F(s,n)= 1/((s-1)n+1) binom{sn}{n}. We find all $n$ such that for $p$ prime, p^q divides F(p^q,n), q>=1. As a byproduct we settle a question of Hough and the late Simion on the…

A elementary proof of Fermat"s Last Theorem[1] is presented for the case of even exponents n=2q, where q is any integer, including 2. For even exponents, the proof of the theorem reduces to showing that solutions of the Pythagorean equation…

Let $p$ and $q$ be distinct primes such that $q+1 | p-1$. In this paper we find all integer solutions $a$, $b$ to the equation $1/a + 1/b = (q+1)/pq$ using only elementary methods.

A polynomial $A(q)=\sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0\le a_1\le \cdots \le a_k\ge a_{k+1} \ge \cdots \ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)=…

We give a method to solve generalized Fermat equations of type $x^4 + y^4 = q z^p$, for some prime values of $q$ and every prime $p$ bigger than 13. We illustrate the method by proving that there are no solutions for $q= 73, 89$ and 113.

In [Acta Math. Univ. Comenianae Vol. LXXX, 1 (2011), pp. 63--70], Yang, Chen and Shi examined the system of difference equations \[ x_n=\frac{a}{y_{n-p}},\qquad y_n=\frac{by_{n-p}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots, \] where $q$ is a…

For every prime number p and integer $n>1$, a simple, involutive, non-degenerate set-theoretic solution $(X,r$) of the Yang-Baxter equation of cardinality $|X| = p^n$ is constructed. Furthermore, for every non-(square-free) positive integer…

We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the integer…

We provide an explicit formulation for the solution to the Catalan's triangle system using Catalan's trapezoids and a specified boundary condition. Additionally, we study this system with various boundary conditions obtained by utilizing…

The classical Pell equation can be extended to the cubic case considering the elements of norm one in $Z[\sqrt[3]{r}]$, which satisfy $x^3 + r y^3 + r^2 z^3 - 3 r x y z = 1$. The solution of the cubic Pell equation is harder than the…

In an earlier paper, Tatong and Suvarnamani explores the Diophantine equation $p^x + p^y = z^2$ for a prime number $p$. In that paper they find some solutions to the equation for $p=2, 3$. In this paper, we look at a general version of this…

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…

Fix any two numbers $p$ and $q$, with $1<p<q$; we give an example of an integral functional enjoying uniform ellipticity and $p$-$q$ growth.

We consider the simultaneous Pell equations $$x^2 - ay^2 = 1, \qquad z^2 - bx^2 = 1,$$ where $a > b\geq 2$ are positive integers. We describe a procedure which, for any fixed $b$, either confirms that the simultaneous Pell equations have at…

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$, $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$. In this article, I…

We prove that if $p \equiv 4,7 \pmod{9}$ is prime and $3$ is not a cube modulo $p$, then both of the equations $x^3+y^3=p$ and $x^3+y^3=p^2$ have a solution with $x,y \in \mathbb{Q}$.

Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and $\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$. Our main aim…

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.