Related papers: Computing rational solutions to $Ax^p+By^p+Cz^p=0$

In this article we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s + A^*X^{-t}A + B^*X^{-p}B = Q$, where $ s, t, p \geq 1$, $ A,…

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd…

Let $E$ be an elliptic curve described by either an Edwards model or a twisted Edwards model over $\mathbb{F}_p$, namely, $E$ is defined by one of the following equations $x^2+y^2=a^2(1+x^2y^2),\, a^5-a\not\equiv 0$ mod $p$, or,…

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…

Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context…

This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider…

Let $E$ be an elliptic curve over $\mathbb{Q}$ described by $y^2= x^3+ Kx+ L$ where $K, L \in \mathbb{Q}$. A set of rational points $(x_i,y_i) \in E(\mathbb{Q})$ for $i=1, 2, \cdots, k$, is said to be a sequence of consecutive cubes on $E$…

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.

We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for…

This paper presents a solution to the following open problem in Number Theory and Geometry: How many points can you find on the (half) parabola $y=x^2$, $x>0$, so that the distance between any pair of them is rational? This problem sounds…

In this paper we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this…

The subject matter of this work is the set of integral points(i.e. points with both coordinates integers) on the graphs of rational functions of the form f(x)=(x^2+bx+c)/(x+a), with a,b,c,being integers.Following the introduction, we…

One of Legendre's theorems on the Diophantine equation $ax^2+by^2+cz^2=0$ provides necessary and sufficient conditions on the existence of nonzero rational solutions of this equation, which helps determine the existence of rational points…

The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…

In our notice we propose the classification of some quartic equations with only trivial solutions by the auxiliary equations. For proving trivial solutions of the quartic equations we use method infinite descent based on the number of prime…

It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$, then there is an approximate derivative at $x_0$. A similar result is…

We prove that for given integers b and c, the diophantine equation x^2+bx+c=y^2, has finitely many integer solutions(i.e. pairs in ZxZ),in fact an even number of such solutions(including the zero or no solutions case).We also offer an…

Let $D$ be a positive nonsquare integer, $p$ a prime number with $p \nmid D$, and $0< \sigma < 0.847$. We show that if the equation $x^2+D=p^n$ has a huge solution $(x_0,n_0)_{(p,\sigma)}$, then there exists an effectively computable…

Let $p\geq 3$ be a prime number. A Fermat curve over $\mathbb{Q}$ of exponent $p$ is defined by an equation of the shape $ax^p+by^p+cz^p=0$, where $a,b,c$ are non-zero rational numbers. We prove in this article that there exist infinitely…

Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve…