Related papers: Rational Points on Erdos-Selfridge Superelliptic C…

For the superelliptic curves of the form $$ (x+1) \cdots(x+i-1)(x+i+1)\cdots (x+k)=y^\ell$$ with $x,y \in \mathbb{Q}$, $y\neq 0$, $k \geq 3$, $1\leq i\leq k$, $\ell \geq 2,$ a prime, Das, Laishram, Saradha, and Edis showed that the…

We study rational points on the Erd\H{o}s-Selfridge curves \begin{align*} y^\ell = x(x+1)\cdots (x+k-1), \end{align*} where $k,\ell\geq 2$ are integers. These curves contain "trivial" rational points $(x,y)$ with $y=0$, and a conjecture of…

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x,y) in K[x,y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f+c) has at most two rational elements. When…

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

In this paper, we study the number of $\mathbb F_{q^n}$-rational points on the affine curve $\mathcal{X}_{d,a,b}$ given by the equation $$ y^d=ax\text{Tr}(x)+b,$$ where $\text{Tr}$ denote the trace function from $\mathbb F_{q^n}$ to…

We study the problem of determining, given an integer $k$, the rational solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$, the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to three elliptic curves…

We show under the assumption that the Tate-Shafarevich group of any elliptic curve over the rational numbers is finite that the cubic surface $x_1^3 + p_1p_2x_2^3 + p_2p_3x_3^3 + p_3p_1x_4^3 = 0$ has a rational point, where $p_1, p_2$ and…

We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3.…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic…

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$…

We look at the elliptic curve E(q), where q is a fixed rational number. A point (p,r) on E(q) is called a rational point if both p and r are rational numbers. We introduce the concept of conjugate points and show that not both can be…

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is…

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 $E$…

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some…

A rational face cuboid is a cuboid that all of edges, two of three face diagonals and space diagonal have rational lengths. \[ E_{1,s}: y^2=x(x-(2s)^2)(x+(s^2-1)^2) \] for a rational number $s \neq 0, \pm 1$, and define $\tilde{A}$…

We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any…

Let $K$ be a field, $a, b\in K$ and $ab\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\geq 2$ the hypersurface given by the…

Let $E$ be an elliptic curve defined over a number field $k$ and $\ell$ a prime integer. When $E$ has at least one $k$-rational point of exact order $\ell$, we derive a uniform upper bound $\exp(C \log B / \log \log B)$ for the number of…

Given a set $S$ of elements in a number field $k$, we discuss the existence of planar algebraic curves over $k$ which possess rational points whose $x$-coordinates are exactly the elements of $S$. If the size $|S|$ of $S$ is either $4,5$,…

In this paper I demonstrate that any pair (m, n) of non-zero and distinct rational numbers may have, at most, four representations as the product of two rational factors such that the sum of factors of m coincides with the sum of factors of…