{"title":"Differences between perfect powers: prime power gaps","authors":"Michael A. Bennett, Samir Siksek","doi":"10.2140/ant.2023.17.1789","DOIUrl":null,"url":null,"abstract":"<p>We develop machinery to explicitly determine, in many instances, when the difference <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\n<mo>−</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math> 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 <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-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 </p>\n<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n<msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\n<mo>+</mo> <msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup>\n<mo>=</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo>\n</math>\n</div>\n<p> where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn>\n<mo>≤</mo>\n<mi>q</mi>\n<mo><</mo> <mn>1</mn><mn>0</mn><mn>0</mn></math> is prime, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>α</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> are integers with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\n<mo>≥</mo> <mn>3</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> gcd</mi><mo> <!--FUNCTION APPLICATION--> </mo><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo>\n<mo>=</mo> <mn>1</mn></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 25","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2023.17.1789","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
We develop machinery to explicitly determine, in many instances, when the difference 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 -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
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.
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