{"title":"Existence of the solutions to the Brocard–Ramanujan problem for norm forms","authors":"Wataru Takeda","doi":"10.1090/bproc/181","DOIUrl":null,"url":null,"abstract":"<p>The Brocard–Ramanujan problem, which is an unsolved problem in number theory, is to find integer solutions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma script l right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(x,\\ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x squared minus 1 equals script l factorial\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>=</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">x^2-1=\\ell !</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Many analogs of this problem are currently being considered. As one example, it is known that there are at most only finitely many algebraic integer solutions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma script l right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(x, \\ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, up to a unit factor, to the equations <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper K Baseline left-parenthesis x right-parenthesis equals script l factorial\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">N_K(x) = \\ell !</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper K\"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">N_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are the norms of number fields <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K slash bold upper Q\"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"bold\">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">K/\\mathbf Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, we construct infinitely many number fields <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper K Baseline left-parenthesis x right-parenthesis equals script l factorial\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">N_K(x) = \\ell !</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has at least <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"22\"> <mml:semantics> <mml:mn>22</mml:mn> <mml:annotation encoding=\"application/x-tex\">22</mml:annotation> </mml:semantics> </mml:math> </inline-formula> solutions for positive integers <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l\"> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"34 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/181","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The Brocard–Ramanujan problem, which is an unsolved problem in number theory, is to find integer solutions (x,ℓ)(x,\ell ) of x2−1=ℓ!x^2-1=\ell !. Many analogs of this problem are currently being considered. As one example, it is known that there are at most only finitely many algebraic integer solutions (x,ℓ)(x, \ell ), up to a unit factor, to the equations NK(x)=ℓ!N_K(x) = \ell !, where NKN_K are the norms of number fields K/QK/\mathbf Q. In this paper, we construct infinitely many number fields KK such that NK(x)=ℓ!N_K(x) = \ell ! has at least 2222 solutions for positive integers ℓ\ell.
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