Pub Date : 2026-03-01Epub Date: 2025-09-05DOI: 10.1016/j.jnt.2025.08.008
Francesco Tropeano
We study monodromy groups associated with elliptic schemes, examining the action induced by the fundamental group of the base via analytic continuation. We develop effective methods for investigating the relative monodromy group of elliptic logarithms and present explicit constructions of loops that simultaneously have trivial action on periods and non-trivial action on logarithms. We provide a new proof that the relative monodromy group of non-torsion sections has full rank. Our results include topological methods and effective techniques for analyzing the ramification locus of sections.
{"title":"Monodromy of elliptic logarithms: Some topological methods and effective results","authors":"Francesco Tropeano","doi":"10.1016/j.jnt.2025.08.008","DOIUrl":"10.1016/j.jnt.2025.08.008","url":null,"abstract":"<div><div>We study monodromy groups associated with elliptic schemes, examining the action induced by the fundamental group of the base via analytic continuation. We develop effective methods for investigating the relative monodromy group of elliptic logarithms and present explicit constructions of loops that simultaneously have trivial action on periods and non-trivial action on logarithms. We provide a new proof that the relative monodromy group of non-torsion sections has full rank. Our results include topological methods and effective techniques for analyzing the ramification locus of sections.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 49-87"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-09-04DOI: 10.1016/j.jnt.2025.08.004
Petr Kucheriaviy
<div><div>A set of natural numbers <em>A</em> is called primitive if no element of <em>A</em> divides any other. Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of prime divisors of <em>n</em> counted with multiplicity. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>z</mi></mrow></msup></mrow></mfrac></math></span>, where <span><math><mi>z</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>. Erdős proved in 1935 that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mi>log</mi><mo></mo><mi>a</mi></mrow></mfrac></math></span> is uniformly bounded over all primitive sets <em>A</em>. We prove a generalization of Erdős inequality which provides an analogous result for <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, when <span><math><mi>z</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Furthermore, we study the supremum of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> over all primitive sets. We also discuss <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mo></mo><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is a generalization of Dirichlet density. We study the asymptotics of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>n</mi><mo>:</mo><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. For <span><math><mi>z</mi><mo>=</mo><mn>1</mn></math></span> we find the next term in asymptotic expansion of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>a</mi></math></span> over all primitive subsets of <span><math><mo>[</mo><mn>1
如果自然数A中的任何元素都不能整除其他自然数，则称自然数A为本数。设Ω(n)为n具有多重性的质因数个数。设fz(A)=∑A∈AzΩ(A) A (log (A) z，其中z∈R>；0。Erdős在1935年证明了f1(A)=∑A∈A1alog (A)在所有原始集A上是一致有界的。我们证明了Erdős不等式的一个推广，对于z∈(0,2)时的fz(A)提供了一个类似的结果。进一步，我们研究了fz(A)在所有原始集合上的最优性。我们还讨论了limz→0 (A)，它是Dirichlet密度的推广。我们研究了fz（Pk）的渐近性，其中Pk={n：Ω(n)=k}。对于z=1，我们找到f1（Pk）的渐近展开中的下一项，改进了Gorodetsky， Lichtman和Wong的结果。我们还研究了∑a∈AzΩ(a)/a在[1，N]的所有原始子集上的最优性。
{"title":"Erdős inequality for primitive sets","authors":"Petr Kucheriaviy","doi":"10.1016/j.jnt.2025.08.004","DOIUrl":"10.1016/j.jnt.2025.08.004","url":null,"abstract":"<div><div>A set of natural numbers <em>A</em> is called primitive if no element of <em>A</em> divides any other. Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of prime divisors of <em>n</em> counted with multiplicity. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>z</mi></mrow></msup></mrow></mfrac></math></span>, where <span><math><mi>z</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>. Erdős proved in 1935 that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mi>log</mi><mo></mo><mi>a</mi></mrow></mfrac></math></span> is uniformly bounded over all primitive sets <em>A</em>. We prove a generalization of Erdős inequality which provides an analogous result for <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, when <span><math><mi>z</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Furthermore, we study the supremum of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> over all primitive sets. We also discuss <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mo></mo><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is a generalization of Dirichlet density. We study the asymptotics of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>n</mi><mo>:</mo><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. For <span><math><mi>z</mi><mo>=</mo><mn>1</mn></math></span> we find the next term in asymptotic expansion of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>a</mi></math></span> over all primitive subsets of <span><math><mo>[</mo><mn>1","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 113-152"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-09-04DOI: 10.1016/j.jnt.2025.08.003
Paul M. Voutier
We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, , arising from the solutions of generalised negative Pell equations, , where −c and are any positive squares. We show that there are at most 2 distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most 5 distinct squares when for infinitely many values of b, including all , as well as once d exceeds an explicit lower bound, without any conditions on the size of such squares.
{"title":"Bounds on the number of squares in recurrence sequences: y0 = b2 (I)","authors":"Paul M. Voutier","doi":"10.1016/j.jnt.2025.08.003","DOIUrl":"10.1016/j.jnt.2025.08.003","url":null,"abstract":"<div><div>We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mo>∞</mo></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span>, arising from the solutions of generalised negative Pell equations, <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>d</mi><msup><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>c</mi></math></span>, where −<em>c</em> and <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are any positive squares. We show that there are at most 2 distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most 5 distinct squares when <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for infinitely many values of <em>b</em>, including all <span><math><mn>1</mn><mo>≤</mo><mi>b</mi><mo>≤</mo><mn>24</mn></math></span>, as well as once <em>d</em> exceeds an explicit lower bound, without any conditions on the size of such squares.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 246-270"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-10-02DOI: 10.1016/j.jnt.2025.09.005
Attila Bérczes , Lajos Hajdu , Florian Luca , István Pink
In this paper, we look at terms of Lucas sequences whose prime factors have indices with bounded gaps in the sequence of all prime numbers. Some of our results depend on certain widely believed conjectures. In our proofs we combine various tools, including Baker's method, the subspace theorem, and results of Stewart, and Murty and Wong.
{"title":"On members of Lucas sequences with bounded prime gaps","authors":"Attila Bérczes , Lajos Hajdu , Florian Luca , István Pink","doi":"10.1016/j.jnt.2025.09.005","DOIUrl":"10.1016/j.jnt.2025.09.005","url":null,"abstract":"<div><div>In this paper, we look at terms of Lucas sequences whose prime factors have indices with bounded gaps in the sequence of all prime numbers. Some of our results depend on certain widely believed conjectures. In our proofs we combine various tools, including Baker's method, the subspace theorem, and results of Stewart, and Murty and Wong.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 897-917"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145262237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-10-01DOI: 10.1016/j.jnt.2025.09.002
Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin , Gyucheol Shin
For a positive integer N, we define an extended modular function of level N motivated by physics and investigate its fundamental properties. Let K be an imaginary quadratic field, and let be an order in K of discriminant D. Let denote the ray class field of modulo . For , we provide an explicit description of the Galois group using special values of extended modular functions of level N and the definite form class group of discriminant D and level N.
{"title":"Extended modular functions and definite form class groups","authors":"Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin , Gyucheol Shin","doi":"10.1016/j.jnt.2025.09.002","DOIUrl":"10.1016/j.jnt.2025.09.002","url":null,"abstract":"<div><div>For a positive integer <em>N</em>, we define an extended modular function of level <em>N</em> motivated by physics and investigate its fundamental properties. Let <em>K</em> be an imaginary quadratic field, and let <span><math><mi>O</mi></math></span> be an order in <em>K</em> of discriminant <em>D</em>. Let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>O</mi><mo>,</mo><mspace></mspace><mi>N</mi></mrow></msub></math></span> denote the ray class field of <span><math><mi>O</mi></math></span> modulo <span><math><mi>N</mi><mi>O</mi></math></span>. For <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span>, we provide an explicit description of the Galois group <span><math><mrow><mi>Gal</mi></mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>O</mi><mo>,</mo><mspace></mspace><mi>N</mi></mrow></msub><mo>/</mo><mi>Q</mi><mo>)</mo></math></span> using special values of extended modular functions of level <em>N</em> and the definite form class group of discriminant <em>D</em> and level <em>N</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 808-824"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-10-01DOI: 10.1016/j.jnt.2025.09.006
Zhigang Tian , Lulu Fang
In this paper, we study two problems concerning the relationship between regular continued fractions (RCFs) and backward continued fractions (BCFs). The first problem addresses Lochs-type theorems for RCFs and BCFs, where we compare the number of partial quotients in one expansion as a function of the number of partial quotients in the other expansion. The second problem investigates the approximation properties of RCFs and BCFs, with particular attention to the set of irrational numbers that are infinitely often better approximated by BCFs than by RCFs. We show that this set has Lebesgue measure zero and further analyze it from the perspectives of Baire category and fractal dimension.
{"title":"Comparing regular and backward continued fractions: Lochs-type theorems and approximation properties","authors":"Zhigang Tian , Lulu Fang","doi":"10.1016/j.jnt.2025.09.006","DOIUrl":"10.1016/j.jnt.2025.09.006","url":null,"abstract":"<div><div>In this paper, we study two problems concerning the relationship between regular continued fractions (RCFs) and backward continued fractions (BCFs). The first problem addresses Lochs-type theorems for RCFs and BCFs, where we compare the number of partial quotients in one expansion as a function of the number of partial quotients in the other expansion. The second problem investigates the approximation properties of RCFs and BCFs, with particular attention to the set of irrational numbers that are infinitely often better approximated by BCFs than by RCFs. We show that this set has Lebesgue measure zero and further analyze it from the perspectives of Baire category and fractal dimension.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 947-972"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-09-04DOI: 10.1016/j.jnt.2025.08.005
Noy Soffer Aranov
We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the random variable which returns the degree of the smallest denominator Q, for which the ball of a fixed radius around a point contains a rational function of the form . Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and P-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.
{"title":"On the minimal denominator problem in function fields","authors":"Noy Soffer Aranov","doi":"10.1016/j.jnt.2025.08.005","DOIUrl":"10.1016/j.jnt.2025.08.005","url":null,"abstract":"<div><div>We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the random variable which returns the degree of the smallest denominator <em>Q</em>, for which the ball of a fixed radius around a point contains a rational function of the form <span><math><mfrac><mrow><mi>P</mi></mrow><mrow><mi>Q</mi></mrow></mfrac></math></span>. Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and <em>P</em>-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 35-48"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-09-23DOI: 10.1016/j.jnt.2025.08.013
Garo Sarajian , Andreas Weingartner
The i-tuply y-densely divisible numbers were introduced by a Polymath project, as a weaker condition on the moduli than y-smoothness, in distribution estimates for primes in arithmetic progressions. We obtain the order of magnitude of the count of these integers up to x, uniformly in x and y, for every fixed natural number i.
{"title":"An extension of smooth numbers: Multiple dense divisibility","authors":"Garo Sarajian , Andreas Weingartner","doi":"10.1016/j.jnt.2025.08.013","DOIUrl":"10.1016/j.jnt.2025.08.013","url":null,"abstract":"<div><div>The <em>i</em>-tuply <em>y</em>-densely divisible numbers were introduced by a Polymath project, as a weaker condition on the moduli than <em>y</em>-smoothness, in distribution estimates for primes in arithmetic progressions. We obtain the order of magnitude of the count of these integers up to <em>x</em>, uniformly in <em>x</em> and <em>y</em>, for every fixed natural number <em>i</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 278-317"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-07-21DOI: 10.1016/j.jnt.2025.06.005
Ramla Abdellatif , Supriya Pisolkar
We prove that the Galois group of the maximal everywhere unramified pro-3-extension L of the biquadratic field has no infinite p-adic analytic pro-3 quotient. This answers negatively a question asked by Boston in his fundamental 1992 paper [4], in which it was observed that the Galois group of , if admitting such a quotient, may provide a counter example to the Fontaine-Mazur conjecture 1.1.
{"title":"Towards the Fontaine-Mazur conjecture for biquadratic extensions: An example","authors":"Ramla Abdellatif , Supriya Pisolkar","doi":"10.1016/j.jnt.2025.06.005","DOIUrl":"10.1016/j.jnt.2025.06.005","url":null,"abstract":"<div><div>We prove that the Galois group of the maximal everywhere unramified pro-3-extension <em>L</em> of the biquadratic field <span><math><mi>K</mi><mo>:</mo><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>26</mn></mrow></msqrt><mo>,</mo><msqrt><mrow><mn>229</mn></mrow></msqrt><mo>)</mo></math></span> has no infinite <em>p</em>-adic analytic pro-3 quotient. This answers negatively a question asked by Boston in his fundamental 1992 paper <span><span>[4]</span></span>, in which it was observed that the Galois group of <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span>, if admitting such a quotient, may provide a counter example to the Fontaine-Mazur conjecture 1.1.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 457-478"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-06-16DOI: 10.1016/j.jnt.2025.03.015
Ndeye Coumba Sarr
A fundamental result of Bass-Serre theory is the following theorem: an abstract group is a free product with amalgamation if and only if it acts on a tree with a segment as fundamental domain. In this article, an analogous result for profinite groups will be given, using the theory of prographs of Deschamps and Suarez introduced in [DSA11].
{"title":"Groupes profinis presqu'amalgamés","authors":"Ndeye Coumba Sarr","doi":"10.1016/j.jnt.2025.03.015","DOIUrl":"10.1016/j.jnt.2025.03.015","url":null,"abstract":"<div><div>A fundamental result of Bass-Serre theory is the following theorem: an abstract group is a free product with amalgamation if and only if it acts on a tree with a segment as fundamental domain. In this article, an analogous result for profinite groups will be given, using the theory of prographs of Deschamps and Suarez introduced in <span><span>[DSA11]</span></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 56-77"},"PeriodicalIF":0.6,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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