Related papers: On the Diophantine Equation X2+19M=YN

Let $n$ be a positive integer. We discuss pairs of distinct odd primes $p$ and $q$ not dividing $n$ for which the Diophantine equations $pq=x^2+ny^2$ have integer solutions in $x$ and $y$. As its examples we classify all such pairs of $p$…

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.

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 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.

The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for…

In this note we investigate the set $S(n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of…

In this paper, we show that the equation $\varphi(|x^{m}-y^{m}|)=|x^{n}-y^{n}|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\neq0, m>0, n>0$ except for the solutions $(x,y,m,n)=((2^{t-1}\pm1),-(2^{t-1}\mp1),2,1),…

This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to…

In this paper we investigate Diophantine equations of the form $T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if…

First, we consider the equation $ax^2 - by^2 + c = 0$, with $a,b \in N*$ and $c \in Z*$, which is a generalization of Pell's equation. Here, we show that: if this equation has an integer solution and $ab$ is not a perfect square, then it…

In this paper, we give a specific way of describing positive integer solutions of a Diophantine equation $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$ and introduce a generalized cluster pattern behind it.

We find all positive integer solutions in $x, y$ and $n$ of $x^2+19^{2k+1}=4y^{n}$ for any non-negative integer $k$.

The quadruple $(1\,484\,801, 1\,203\,120, 1\,169\,407, 1\,157\,520)$ already known is essentially the only non-trivial solution of the Diophantine equation $x^4 + 2 y^4 = z^4 + 4 w^4$ for $|x|$, $|y|$, $|z|$, and $|w|$ up to one hundred…

We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.

In this short note we study the existence and number of solutions in the set of integers ($Z$) and in the set of natural numbers ($N$) of Diopahntine Equations of second degree with two variables of the general form $ax^2-by^2=c$.

This paper is a continuation of [1], in which I studied Harvey Friedman's problem of whether the function f(x,y) = x^2 + y^3 satisfies any identities; however, no knowledge of [1] is necessary to understand this paper. We will break the…

In the following we consider Diophantine equations of the form $x^2+ zxy + y^2 = M$ for given $M,z \in \mathbb{Z}$ and discuss the number of its (primitive) solutions as well as the construction of them. To reach this goal we introduce…

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm g(y)^2$ for some simple Laurent polynomials $f$ and $g$.

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…

The Diophantine equation (x^n-1)/(x-1)=y^q has four known solutions in integers x, y, q and n with |x|, |y|, q > 1 and n > 2. Whilst we expect that there are, in fact, no more solutions, such a result is well beyond current technology. In…