Related papers: Diophantus Equations and Partially Ordered Sets

Let $k,\ell\geq2$ be fixed integers and $C$ be an effectively computable constant depending only on $k$ and $\ell$. In this paper, we prove that all solutions of the equation $(x+1)^{k}+(x+2)^{k}+...+(\ell x)^{k}=y^{n}$ in integers $x,y,n$…

In this note we will analyze a diophantine equation raised by Michael Bennett in [1] that is pivotal in establishing that powers of five has few digits in its ternary expansion. We will show that the Diophantine equation…

We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions…

In this paper one shows if the number of natural solutions of a general linear equation is limited or not. Also, it is presented a method of solving the Diophantine equation $ax-by=c$ in the set of natural numbers, and an example of solving…

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type…

The object of this paper is to give a new proof of all the solutions of the Diophantine equation x^2+11^m=y^n; in positive integers x, y with odd m>1 and n>=3.

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.

By means of elementary conditions on coefficients, we isolate a large class of Fermat-like Diophantine equations that are not partition regular, the simplest examples being $x^n+y^m=z^k$ with $k\notin\{n,m\}$.

We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…

We investigate a family of Diophantine polynomial equations which involve continuant functions. In particular, given a polynomial $P(x)\in \mathbb{Z}[x]$ and $n\in \mathbb{N}$, we consider the equation $P(K_n(x_1,\ldots, x_n)) =…

Naciri proved that for any integer $k\geq2$, the Brocard--Ramanujan equation $n!+1=x^2$ has only finitely many integer solutions, assuming $x\pm1$ is a $k$-free integer or a prime power. In the present paper we prove similar statements for…

In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known…

Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive…

We show that if $k\ge 2$ is an integer and $(F_n^{(k)})_{n\ge 0}$ is the sequence of $k$-generalized Fibonacci numbers, then there are only finitely many triples of positive integers $1<a<b<c$ such that $ab+1,~ac+1,~bc+1$ are all members of…

We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…

By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and…

We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…

Let $k$ be a positive integer, and let $a,b$ be coprime positive integers with $\min\{a,b\}>1$. In this paper, using a combination of some elementary number theory techniques with classical results on the Nagell-Ljunggren equation, the…

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.

Motivated by the recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of…