完美权力之间的差异：主要权力差距

IF 0.9 1区数学 Q2 MATHEMATICS

Algebra & Number Theory Pub Date : 2023-09-19 DOI:10.2140/ant.2023.17.1789

Michael A. Bennett, Samir Siksek

引用次数: 4

摘要

我们开发了一种机制，在许多情况下，明确地确定差值x2−yn何时只能被给定固定素数的幂整除。这结合了Diophantine近似（对数线性形式的边界，包括阿基米德和非阿基米德，格基约简，求解Thue–Mahler和S单元方程的方法，以及Bilu、Hanrot和Voutier的原始除数定理）和经典代数数论的各种技术，其结果来自附加到Frey–Hellegourch椭圆曲线的Galois表示的模块性。举例来说，我们完全求解方程x2+qα=yn，其中2≤q<；100是素数，x、y、α和n是n≥3且gcd的整数⁡ （x，y）=1。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Differences between perfect powers: prime power gaps

We develop machinery to explicitly determine, in many instances, when the difference $x^{2} - y^{n}$ is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and $S$ -unit equations, and the primitive divisor theorem of Bilu, Hanrot and Voutier) and classical algebraic number theory, with results derived from the modularity of Galois representations attached to Frey–Hellegoaurch elliptic curves. By way of example, we completely solve the equation

x^{2} + q^{α} = y^{n},

where $2 \leq q < 100$ is prime, and $x, y, α$ and $n$ are integers with $n \geq 3$ and $\gcd (x, y) = 1$ .

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来源期刊

Algebra & Number Theory MATHEMATICS-

CiteScore

1.80

自引率

7.70%

发文量

审稿时长

6-12 weeks

期刊介绍： ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.