Related papers: There is no Diophantine quintuple

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers…

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…

In 2012, T. Miyazaki and A. Togb\'{e} gave all of the solutions of the Diophantine equations $(2am-1)^x+(2m)^y=(2am+1)^z$ and $b^x+2^y=(b+2)^z$ in positive integers $x,y,z,$ $a>1$ and $b\ge 5$ odd. In this paper, we propose a similar…

Crux Mathematicorum with Mathematical Mayhem, is a problem solving journal published by the Canadian Mathematical Society. In the March 2010 issue(see reference[1]) ,the following problem was proposed:Determine all positive integers a,b,…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) the function f is…

Let $\lambda_i, \mu_j$ be non-zero real numbers not all of the same sign and let $a_i, b_k$ be non-zero integers not all of the same sign. We investigate a mixed Diophantine system of the shape \begin{equation*} \begin{cases} \left|…

Let $n$ be a positive integer. The Diophantine equation $n(x_1+x_2+\dots +x_n)=x_1x_2\dots x_n$, $1 \le x_1\le x_2\le \dots \le x_n$ is called Erd\H{o}s's last equation. We prove that $x_n\to \infty $ as $n\to \infty$ and determine all…

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^\alpha C^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for "most" primes $q$ and suitably large prime…

In this paper, we consider the equations involving Euler's totient function $\phi$ and Lucas type sequences. In particular, we prove that the equation $\phi (x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for…

Let $f(x)=x^{2}(x^{2}-1)(x^{2}-2)(x^{2}-3).$ We prove that the Diophantine equation $ f(x)=2f(y)$ has no solutions in positive integers $x$ and $y$, except $(x, y)=(1, 1)$.

In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups…

A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence…

We show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers. We also prove that there is a diophantine model of the…

We prove that the Diophantine equation (a^2 c x^k - 1)(b^2 c y^k - 1) = (abc z^k - 1)^2 has no solutions in positive integers with x, y, z > 1, k \geq 7 and a^2x^k \neq b^2y^k.

Let $\gamma\in(0;\frac{1}{2}),\tau\geq 1$ and define the "$\gamma,\tau$ Diophantine set" as: $$D_{\gamma,\tau}:=\{\alpha\in (0;1): ||q\alpha||\geq\frac{\gamma}{q^{\tau}}\quad\forall q\in\Bbb{N}\},\qquad||x||:=\inf_{p\in\Bbb{Z}}|x-p|. $$ In…

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…

Fix a positive real number $\theta$. The natural numbers $m$ with largest square-free divisor not exceeding $m^\theta$ form a set $\mathscr{A}$, say. It is shown that whenever $\theta>1/2$ then all large natural numbers $n$ are the sum of…

Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple.

In this paper, we study properties of the Diophantine exponents $w_n$ and $w_n^{*}$ for Laurent series over a finite field. We prove that for an integer $n\geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of…

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ \theta ,$ where $ \theta >2$ is real and non-integral. For fixed non-zero real numbers $ \lambda_i $ not all of the same sign we write…