scholarly journals RATIONAL POINTS ON THREE SUPERELLIPTIC CURVES

2011 ◽  
Vol 85 (1) ◽  
pp. 105-113 ◽  
Author(s):  
ZHONGYAN SHEN ◽  
TIANXIN CAI
Keyword(s):  
Rational Points ◽  
Superelliptic Curves ◽  
Image Position

AbstractIn this paper, we obtain all rational points (x,y) on the superelliptic curves

scholarly journals Rational points on Erdős–Selfridge superelliptic curves

2016 ◽  
Vol 152 (11) ◽  
pp. 2249-2254 ◽  
Author(s):  
Michael A. Bennett ◽  
Samir Siksek
Keyword(s):  
Rational Numbers ◽  
Rational Points ◽  
Superelliptic Curves ◽  
Image Position

Given $k\geqslant 2$, we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$) for which $$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

Introduction to the Schwartz Space of T\G

1989 ◽  
Vol 41 (2) ◽  
pp. 285-320 ◽  
Author(s):  
W. Casselman
Keyword(s):  
Reductive Group ◽  
Rational Points ◽  
Schwartz Space ◽  
Parabolic Subgroups ◽  
Arithmetic Groups ◽  
Similar Question ◽  
Arithmetic Subgroup ◽  
Cohomology Of Arithmetic Groups ◽  
Image Position

Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G. This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

VARIANTS OF ERDŐS–SELFRIDGE SUPERELLIPTIC CURVES AND THEIR RATIONAL POINTS

Mathematika ◽  
2018 ◽  
Vol 64 (2) ◽  
pp. 380-386 ◽  
Author(s):  
Pranabesh Das ◽  
Shanta Laishram ◽  
N. Saradha
Keyword(s):  
Rational Points ◽  
Superelliptic Curves

Simultaneous diophantine approximations and Hermite's method

1980 ◽  
Vol 21 (3) ◽  
pp. 463-470 ◽  
Author(s):  
Alain Durand
Keyword(s):  
Exponential Function ◽  
Integer Point ◽  
Rational Numbers ◽  
Rational Points ◽  
Rational Approximations ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations ◽  
Image Position

In this paper we generalize a result of Mahler on rational approximations of the exponential function at rational points by proving the following theorem: let n ε N* and αl, …, αn be distinct non-zero rational numbers; there exists a constant c = c(n, αl, …, αn) ≥ 0 such thatfor every non-zero integer point (qo, ql, …, qn)and q = max {|ql|, … |qn|, 3}.

scholarly journals Note on rational approximations of the exponential function at rational points

1976 ◽  
Vol 14 (3) ◽  
pp. 449-455 ◽  
Author(s):  
Alain Durand
Keyword(s):  
Exponential Function ◽  
Rational Points ◽  
Rational Approximations ◽  
Image Position ◽  
Positive Integers

Let p, q, u, and v be any four positive integers, and let δ be a number in the interval 0 < δ ≤ 2. In one of his papers, Kurt Mahler, Bull. Austral. Math. Soc. 10 (1974), 325–335, proved that if q satisfies the inequalitiesthenIn this note, by a slightly different treatment of some inequalities in Mahler's paper, we easily obtain the same result with q only restricted by the first condition.

The diophantine equation Ax4 + By4 + Cz4 = 0

1970 ◽  
Vol 68 (1) ◽  
pp. 125-128
Author(s):  
L. J. Mordell
Keyword(s):  
Finite Number ◽  
Diophantine Equation ◽  
Rational Points ◽  
Image Position

In a paper (1) published in 1922 in these Proceedings, I conjectured that a curve of genus greater than one contained only a finite number of rational points, but this has never been proved. A particular case is given by the equation of genus three,and so any results for such equations may be of interest.

FROM SEPARABLE POLYNOMIALS TO NONEXISTENCE OF RATIONAL POINTS ON CERTAIN HYPERELLIPTIC CURVES

2014 ◽  
Vol 96 (3) ◽  
pp. 354-385 ◽  
Author(s):  
NGUYEN NGOC DONG QUAN
Keyword(s):  
Hyperelliptic Curves ◽  
Rational Points ◽  
Hasse Principle ◽  
Sufficient Condition ◽  
Separability Criterion ◽  
Image Position

AbstractWe give a separability criterion for the polynomials of the form $$\begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}$$ Using this separability criterion, we prove a sufficient condition using the Brauer–Manin obstruction under which curves of the form $$\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$ have no rational points. As an illustration, using the sufficient condition, we study the arithmetic of hyperelliptic curves of the above form and show that there are infinitely many curves of the above form that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION

2006 ◽  
Vol 92 (2) ◽  
pp. 273-306 ◽  
Author(s):  
M. A. BENNETT ◽  
N. BRUIN ◽  
K. GYÖRY ◽  
L. HAJDU
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Diophantine Equations ◽  
Galois Representations ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Superelliptic Curves

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

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