Pub Date : 2024-01-01Epub Date: 2024-05-13DOI: 10.1007/s40993-024-00534-5
Sebastian Heintze, Volker Ziegler
In this paper, we consider the Diophantine equation for given integers b, c with , whereas varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (n, m) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of .
在本文中，我们考虑了给定整数 b, c 的二阶方程 Vn-bm=c，b≥2，而 Vn 在第二类卢卡斯-雷默序列中变化。我们在一些技术条件下证明，如果所考虑的方程至少有三个解 (n, m) ，那么解的大小以及 Vn 的特征多项式系数的大小都有一个上限。
{"title":"On Pillai's Problem involving Lucas sequences of the second kind.","authors":"Sebastian Heintze, Volker Ziegler","doi":"10.1007/s40993-024-00534-5","DOIUrl":"10.1007/s40993-024-00534-5","url":null,"abstract":"<p><p>In this paper, we consider the Diophantine equation <math><mrow><msub><mi>V</mi><mi>n</mi></msub><mo>-</mo><msup><mi>b</mi><mi>m</mi></msup><mo>=</mo><mi>c</mi></mrow></math> for given integers <i>b</i>, <i>c</i> with <math><mrow><mi>b</mi><mo>≥</mo><mn>2</mn></mrow></math>, whereas <math><msub><mi>V</mi><mi>n</mi></msub></math> varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (<i>n</i>, <i>m</i>) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of <math><msub><mi>V</mi><mi>n</mi></msub></math>.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"10 2","pages":"51"},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11090840/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140945180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-17DOI: 10.1007/s40993-023-00488-0
Abbey Bourdon, David R. Gill, Jeremy Rouse, Lori D. Watson
{"title":"Odd degree isolated points on $$X_1(N)$$ with rational j-invariant","authors":"Abbey Bourdon, David R. Gill, Jeremy Rouse, Lori D. Watson","doi":"10.1007/s40993-023-00488-0","DOIUrl":"https://doi.org/10.1007/s40993-023-00488-0","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"347 2‐3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138966733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-22DOI: 10.1007/s40993-023-00481-7
Cameron Bjorklund, Matthew Litman
{"title":"Error approximation for backwards and simple continued fractions","authors":"Cameron Bjorklund, Matthew Litman","doi":"10.1007/s40993-023-00481-7","DOIUrl":"https://doi.org/10.1007/s40993-023-00481-7","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"55 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139248580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1007/s40993-023-00486-2
Maryam Elaoud, Mohamed Hbaib
{"title":"Transcendence criterion with $$(beta ,{mathcal {A}})$$-representations in some quadratic integer bases","authors":"Maryam Elaoud, Mohamed Hbaib","doi":"10.1007/s40993-023-00486-2","DOIUrl":"https://doi.org/10.1007/s40993-023-00486-2","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"7 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135142137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1007/s40993-022-00402-0
Daniel J. Bernstein
Abstract This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in S -unit searches (for, e.g., class-group computation) is computing a $$Theta (nlog n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -bit norm of an element of weight $$n^{1/2+o(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> in a degree- n field; this method then uses $$n(log n)^{3+o(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> bit operations. An $$n(log n)^{O(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader. As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least $$n^2(log n)^{2+o(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm $
摘要本文给出了一种快速计算光滑分环域的积分元的代数范数的方法，更一般地，给出了具有可交换伽罗瓦群的光滑伽罗瓦数域的代数范数。在S单元搜索(例如，类-组计算)中出现的典型场景是计算权重为$$n^{1/2+o(1)}$$ n 1 / 2 + o(1)的元素的$$Theta (nlog n)$$ Θ (n log n)位范数在度- n字段中;这个方法然后使用$$n(log n)^{3+o(1)}$$ n (log n) 3 + o(1)位运算。$$n(log n)^{O(1)}$$ n (log n) O(1)个运算次数在两种更简单的特殊情况下是已知的:从2次幂的环切域通过2次幂的环切子域的塔的模，以及从多二次域通过多二次子域的塔的模。本文通过识别支持快速乘法的塔兼容积分基来处理更一般的阿贝尔域;特别是，在塔兼容的高斯周期积分基和Rader的快速乘法思想之间存在协同作用。作为基准，本文还分析了适用于任意数字字段的各种标准规范计算技术，得出的结论是，所有这些技术在相同的场景中至少使用$$n^2(log n)^{2+o(1)}$$ n 2 (log n) 2 + o(1)位操作，即使使用用于连分式和复杂fft的快速子程序。与此基线相比，专用于光滑度阿贝尔域的算法查找每个范数的速度要快$$n/(log n)^{1+o(1)}$$ n / (log n) 1 + o(1)倍，并且在S -unit搜索中完成范数计算的速度要快$$n^2/(log n)^{1+o(1)}$$ n 2 / (log n) 1 + o(1)倍。
{"title":"Fast norm computation in smooth-degree Abelian number fields","authors":"Daniel J. Bernstein","doi":"10.1007/s40993-022-00402-0","DOIUrl":"https://doi.org/10.1007/s40993-022-00402-0","url":null,"abstract":"Abstract This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in S -unit searches (for, e.g., class-group computation) is computing a $$Theta (nlog n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -bit norm of an element of weight $$n^{1/2+o(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> in a degree- n field; this method then uses $$n(log n)^{3+o(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> bit operations. An $$n(log n)^{O(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader. As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least $$n^2(log n)^{2+o(1)}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm $","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"116 23","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1007/s40993-023-00484-4
Alexis LaBelle, Emily Van Bergeyk, Matthew P. Young
{"title":"Reciprocity and the kernel of Dedekind sums","authors":"Alexis LaBelle, Emily Van Bergeyk, Matthew P. Young","doi":"10.1007/s40993-023-00484-4","DOIUrl":"https://doi.org/10.1007/s40993-023-00484-4","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":" 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135292694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1007/s40993-023-00480-8
Bernhard Heim, Markus Neuhauser, Robert Tröger
Abstract The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.
D 'Arcais多项式(也称为Nekrasov-Okounkov多项式)的零表示Dedekind函数幂的傅里叶系数的消失。这些多项式满足非常系数遗传型差分方程。我们将D 'Arcais多项式与满足卷积型Volterra差分方程的多项式联系起来。我们得到了关于零位置转移的结果。作为应用，我们得到了第二类切比雪夫多项式与1相关拉盖尔多项式之间的恒等式。我们得到了Lehmer猜想的一个新版本和Hermite多项式的零点界。
{"title":"Zeros transfer for recursively defined polynomials","authors":"Bernhard Heim, Markus Neuhauser, Robert Tröger","doi":"10.1007/s40993-023-00480-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00480-8","url":null,"abstract":"Abstract The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1007/s40993-023-00485-3
Takahiko Fujita, Naohiro Yoshida
{"title":"On further application of the zeta distribution to number theory","authors":"Takahiko Fujita, Naohiro Yoshida","doi":"10.1007/s40993-023-00485-3","DOIUrl":"https://doi.org/10.1007/s40993-023-00485-3","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"27 40","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135391919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.1007/s40993-022-00415-9
Noam D. Elkies, Gaurav Goel
{"title":"On powerful integers expressible as sums of two coprime fourth powers","authors":"Noam D. Elkies, Gaurav Goel","doi":"10.1007/s40993-022-00415-9","DOIUrl":"https://doi.org/10.1007/s40993-022-00415-9","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"52 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135474994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1007/s40993-023-00479-1
Masatake Hirao, Hiroshi Nozaki, Koji Tasaka
{"title":"Spherical designs and modular forms of the $$D_4$$ lattice","authors":"Masatake Hirao, Hiroshi Nozaki, Koji Tasaka","doi":"10.1007/s40993-023-00479-1","DOIUrl":"https://doi.org/10.1007/s40993-023-00479-1","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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